Daily Papers

Large language models often remain sensitive to answer format: a question solved correctly in one form may fail in another semantically equivalent form. To study this gap, we define cross-format robustness as the extent to which a model answers the same underlying question consistently across formats. We then compare full-format training with FormatMix, which expands only a subset of training items into multiple equivalent formats using either random or targeted selection. Across GLM4 and Llama-3.1, multi-format supervision consistently improves both task performance and cross-format robustness, whereas Multiple-choice question (MCQ)-only supervision alone brings little benefit and can even reduce robustness. We further find that expanding only about 30% of the training set into multiple formats often recovers most of the gain from full-format training, and this effect appears across the model families and sizes we study. These results suggest that format diversity, rather than additional supervision alone, is the key driver of robustness. That lightweight multi-format augmentation is a practical way to make LLMs less sensitive to answer format without changing the base model.

A fundamental problem in combinatorial optimization is identifying equivalent formulations, which can lead to more efficient solution strategies and deeper insights into a problem's computational complexity. The need to automatically identify equivalence between problem formulations has grown as optimization copilots--systems that generate problem formulations from natural language descriptions--have proliferated. However, existing approaches to checking formulation equivalence lack grounding, relying on simple heuristics which are insufficient for rigorous validation. Inspired by Karp reductions, in this work we introduce quasi-Karp equivalence, a formal criterion for determining when two optimization formulations are equivalent based on the existence of a mapping between their decision variables. We propose EquivaMap, a framework that leverages large language models to automatically discover such mappings, enabling scalable and reliable equivalence verification. To evaluate our approach, we construct the first open-source dataset of equivalent optimization formulations, generated by applying transformations such as adding slack variables or valid inequalities to existing formulations. Empirically, EquivaMap significantly outperforms existing methods, achieving substantial improvements in correctly identifying formulation equivalence.

Contrastive learning -- a modern approach to extract useful representations from unlabeled data by training models to distinguish similar samples from dissimilar ones -- has driven significant progress in foundation models. In this work, we develop a new theoretical framework for analyzing data augmentation-based contrastive learning, with a focus on SimCLR as a representative example. Our approach is based on the concept of approximate sufficient statistics, which we extend beyond its original definition in oko2025statistical for contrastive language-image pretraining (CLIP) using KL-divergence. We generalize it to equivalent forms and general f-divergences, and show that minimizing SimCLR and other contrastive losses yields encoders that are approximately sufficient. Furthermore, we demonstrate that these near-sufficient encoders can be effectively adapted to downstream regression and classification tasks, with performance depending on their sufficiency and the error induced by data augmentation in contrastive learning. Concrete examples in linear regression and topic classification are provided to illustrate the broad applicability of our results.

This work studies discrete diffusion probabilistic models with applications to natural language generation. We derive an alternative yet equivalent formulation of the sampling from discrete diffusion processes and leverage this insight to develop a family of reparameterized discrete diffusion models. The derived generic framework is highly flexible, offers a fresh perspective of the generation process in discrete diffusion models, and features more effective training and decoding techniques. We conduct extensive experiments to evaluate the text generation capability of our model, demonstrating significant improvements over existing diffusion models.

Crystals are the foundation of numerous scientific and industrial applications. While various learning-based approaches have been proposed for crystal generation, existing methods seldom consider the space group constraint which is crucial in describing the geometry of crystals and closely relevant to many desirable properties. However, considering space group constraint is challenging owing to its diverse and nontrivial forms. In this paper, we reduce the space group constraint into an equivalent formulation that is more tractable to be handcrafted into the generation process. In particular, we translate the space group constraint into two parts: the basis constraint of the invariant logarithmic space of the lattice matrix and the Wyckoff position constraint of the fractional coordinates. Upon the derived constraints, we then propose DiffCSP++, a novel diffusion model that has enhanced a previous work DiffCSP by further taking space group constraint into account. Experiments on several popular datasets verify the benefit of the involvement of the space group constraint, and show that our DiffCSP++ achieves promising performance on crystal structure prediction, ab initio crystal generation and controllable generation with customized space groups.

Bayesian filtering approximates the true underlying behavior of a time-varying system by inverting an explicit generative model to convert noisy measurements into state estimates. This process typically requires either storage, inversion, and multiplication of large matrices or Monte Carlo estimation, neither of which are practical in high-dimensional state spaces such as the weight spaces of artificial neural networks. Here, we frame the standard Bayesian filtering problem as optimization over a time-varying objective. Instead of maintaining matrices for the filtering equations or simulating particles, we specify an optimizer that defines the Bayesian filter implicitly. In the linear-Gaussian setting, we show that every Kalman filter has an equivalent formulation using K steps of gradient descent. In the nonlinear setting, our experiments demonstrate that our framework results in filters that are effective, robust, and scalable to high-dimensional systems, comparing well against the standard toolbox of Bayesian filtering solutions. We suggest that it is easier to fine-tune an optimizer than it is to specify the correct filtering equations, making our framework an attractive option for high-dimensional filtering problems.

Formal proofs are challenging to write even for experienced experts. Recent progress in Neural Theorem Proving (NTP) shows promise in expediting this process. However, the formal corpora available on the Internet are limited compared to the general text, posing a significant data scarcity challenge for NTP. To address this issue, this work proposes Alchemy, a general framework for data synthesis that constructs formal theorems through symbolic mutation. Specifically, for each candidate theorem in Mathlib, we identify all invocable theorems that can be used to rewrite or apply to it. Subsequently, we mutate the candidate theorem by replacing the corresponding term in the statement with its equivalent form or antecedent. As a result, our method increases the number of theorems in Mathlib by an order of magnitude, from 110k to 6M. Furthermore, we perform continual pretraining and supervised finetuning on this augmented corpus for large language models. Experimental results demonstrate the effectiveness of our approach, achieving a 5% absolute performance improvement on Leandojo benchmark. Additionally, our synthetic data achieve a 2.5% absolute performance gain on the out-of-distribution miniF2F benchmark. To provide further insights, we conduct a comprehensive analysis of synthetic data composition and the training paradigm, offering valuable guidance for developing a strong theorem prover.

Large nonlinear recurrent neural networks with random couplings generate high-dimensional, potentially chaotic activity whose structure is of interest in neuroscience and other fields. A fundamental object encoding the collective structure of this activity is the N times N covariance matrix. Prior analytical work on the covariance matrix has been limited to low-dimensional summary statistics. Recent work proposed an ansatz in which, at large N, the covariance matrix for a typical quenched realization takes the same form as that of a linear network with the same couplings, driven by independent noise, with DMFT order parameters setting the transfer function and the noise spectrum. Here, we derive this ansatz using the two-site cavity method, providing two derivations with complementary perspectives. The first decomposes each unit's activity into a linear response to its local field and a nonlinear residual, and shows that cross-covariances between residuals at distinct sites are strongly suppressed, so the residuals act as independent noise driving a linear network. The second derives a self-consistent matrix equation for the covariance matrix. A naive Gaussian closure for the joint statistics of local fields at distinct sites misses cross terms that, in a linear network, would be generated by an external drive. The cavity method recovers these terms from non-Gaussian contributions, revealing an emergent external drive. Higher-order cross-site moments follow a Wick-like decomposition into products of pairwise covariances at leading order, reducing them to the linear-equivalent form. We verify the predictions in simulations. These results extend linear equivalence from feedforward high-dimensional nonlinear systems, where the activations are independent of the weights, to recurrent networks, where the activations are correlated with the couplings that generate them.

Retrieval systems often fail when user queries differ stylistically or semantically from the language used in domain documents. Query rewriting has been proposed to bridge this gap, improving retrieval by reformulating user queries into semantically equivalent forms. However, most existing approaches overlook the stylistic characteristics of target documents-their domain-specific phrasing, tone, and structure-which are crucial for matching real-world data distributions. We introduce a retrieval feedback-driven dataset generation framework that automatically identifies failed retrieval cases, leverages large language models to rewrite queries in the style of relevant documents, and verifies improvement through re-retrieval. The resulting corpus of (original, rewritten) query pairs enables the training of rewriter models that are explicitly aware of document style and retrieval feedback. This work highlights a new direction in data-centric information retrieval, emphasizing how feedback loops and document-style alignment can enhance the reasoning and adaptability of RAG systems in real-world, domain-specific contexts.

Logical reasoning task has attracted great interest since it was proposed. Faced with such a task, current competitive models, even large language models (e.g., ChatGPT and PaLM 2), still perform badly. Previous promising LMs struggle in logical consistency modeling and logical structure perception. To this end, we model the logical reasoning task by transforming each logical sample into reasoning paths and propose an architecture PathReasoner. It addresses the task from the views of both data and model. To expand the diversity of the logical samples, we propose an atom extension strategy supported by equivalent logical formulas, to form new reasoning paths. From the model perspective, we design a stack of transformer-style blocks. In particular, we propose a path-attention module to joint model in-atom and cross-atom relations with the high-order diffusion strategy. Experiments show that PathReasoner achieves competitive performances on two logical reasoning benchmarks and great generalization abilities.

State space models (SSM) have recently been shown to be very effective as a deep learning layer as a promising alternative to sequence models such as RNNs, CNNs, or Transformers. The first version to show this potential was the S4 model, which is particularly effective on tasks involving long-range dependencies by using a prescribed state matrix called the HiPPO matrix. While this has an interpretable mathematical mechanism for modeling long dependencies, it introduces a custom representation and algorithm that can be difficult to implement. On the other hand, a recent variant of S4 called DSS showed that restricting the state matrix to be fully diagonal can still preserve the performance of the original model when using a specific initialization based on approximating S4's matrix. This work seeks to systematically understand how to parameterize and initialize such diagonal state space models. While it follows from classical results that almost all SSMs have an equivalent diagonal form, we show that the initialization is critical for performance. We explain why DSS works mathematically, by showing that the diagonal restriction of S4's matrix surprisingly recovers the same kernel in the limit of infinite state dimension. We also systematically describe various design choices in parameterizing and computing diagonal SSMs, and perform a controlled empirical study ablating the effects of these choices. Our final model S4D is a simple diagonal version of S4 whose kernel computation requires just 2 lines of code and performs comparably to S4 in almost all settings, with state-of-the-art results for image, audio, and medical time-series domains, and averaging 85\% on the Long Range Arena benchmark.

Despite significant advancements in large language models (LLMs) that enhance robot agents' understanding and execution of natural language (NL) commands, ensuring the agents adhere to user-specified constraints remains challenging, particularly for complex commands and long-horizon tasks. To address this challenge, we present three key insights, equivalence voting, constrained decoding, and domain-specific fine-tuning, which significantly enhance LLM planners' capability in handling complex tasks. Equivalence voting ensures consistency by generating and sampling multiple Linear Temporal Logic (LTL) formulas from NL commands, grouping equivalent LTL formulas, and selecting the majority group of formulas as the final LTL formula. Constrained decoding then uses the generated LTL formula to enforce the autoregressive inference of plans, ensuring the generated plans conform to the LTL. Domain-specific fine-tuning customizes LLMs to produce safe and efficient plans within specific task domains. Our approach, Safe Efficient LLM Planner (SELP), combines these insights to create LLM planners to generate plans adhering to user commands with high confidence. We demonstrate the effectiveness and generalizability of SELP across different robot agents and tasks, including drone navigation and robot manipulation. For drone navigation tasks, SELP outperforms state-of-the-art planners by 10.8% in safety rate (i.e., finishing tasks conforming to NL commands) and by 19.8% in plan efficiency. For robot manipulation tasks, SELP achieves 20.4% improvement in safety rate. Our datasets for evaluating NL-to-LTL and robot task planning will be released in github.com/lt-asset/selp.

Logical reasoning is a key task for artificial intelligence due to it's role in major downstream tasks such as Question Answering, Summarization. Recent methods in improving the reasoning ability of LLMs fall short in correctly converting a natural language reasoning problem to an equivalent logical formulation, which hinders the framework's overall ability to reason. Towards this, we propose to use finetuning on a preference optimization dataset to learn to parse and represent a natural language problem as a whole to a consistent logical program by 1) introducing a new supervised and preference optimization dataset LogicPO, and 2) adopting popular techniques such as Direct Preference Optimization (DPO), Kahneman-Tversky optimization (KTO) to finetune open-source LLMs. Our best model with Phi-3.5 consistently outperforms GPT-3.5-turbo's (8-shot) by producing 10% more logically correct and with 14% less syntax errors. Through the framework and our improved evaluation metrics, we offer a promising direction in improving the logical reasoning of LLMs by better representing them in their logical formulations.

Automated theorem proving (ATP) benchmarks largely consist of problems formalized in MathLib, so current ATP training and evaluation are heavily biased toward MathLib's definitional framework. However, frontier mathematics is often exploratory and prototype-heavy, relying on bespoke constructions that deviate from standard libraries. In this work, we evaluate the robustness of current ATP systems when applied to a novel definitional framework, specifically examining the performance gap between standard library problems and bespoke mathematical constructions. We introduce TaoBench, an undergraduate-level benchmark derived from Terence Tao's Analysis I, which formalizes analysis by constructing core mathematical concepts from scratch, without relying on standard Mathlib definitions, as well as by mixing from-scratch and MathLib constructions. For fair evaluation, we build an agentic pipeline that automatically extracts a compilable, self-contained local environment for each problem. To isolate the effect of definitional frameworks, we additionally translate every problem into a mathematically equivalent Mathlib formulation, yielding paired TaoBench-Mathlib statements for direct comparison. While state-of-the-art ATP models perform capably within the MathLib framework, performance drops by an average of roughly 26% on the definitionally equivalent Tao formulation. This indicates that the main bottleneck is limited generalization across definitional frameworks rather than task difficulty. TaoBench thus highlights a gap between benchmark performance and applicability, and provides a concrete foundation for developing and testing provers better aligned with research mathematics.

Deep neural networks have demonstrated remarkable performance across numerous learning tasks but often suffer from miscalibration, resulting in unreliable probability outputs. This has inspired many recent works on mitigating miscalibration, particularly through post-hoc recalibration methods that aim to obtain calibrated probabilities without sacrificing the classification performance of pre-trained models. In this study, we summarize and categorize previous works into three general strategies: intuitively designed methods, binning-based methods, and methods based on formulations of ideal calibration. Through theoretical and practical analysis, we highlight ten common limitations in previous approaches. To address these limitations, we propose a probabilistic learning framework for calibration called h-calibration, which theoretically constructs an equivalent learning formulation for canonical calibration with boundedness. On this basis, we design a simple yet effective post-hoc calibration algorithm. Our method not only overcomes the ten identified limitations but also achieves markedly better performance than traditional methods, as validated by extensive experiments. We further analyze, both theoretically and experimentally, the relationship and advantages of our learning objective compared to traditional proper scoring rule. In summary, our probabilistic framework derives an approximately equivalent differentiable objective for learning error-bounded calibrated probabilities, elucidating the correspondence and convergence properties of computational statistics with respect to theoretical bounds in canonical calibration. The theoretical effectiveness is verified on standard post-hoc calibration benchmarks by achieving state-of-the-art performance. This research offers valuable reference for learning reliable likelihood in related fields.

Existing LLM routing methods typically treat a model's single response to a query as its capability label for training routers. However, because LLM generation is inherently stochastic, such single-shot supervision provides only a noisy observation of a query-model pair's behavior rather than a reliable capability estimate. We show that this assumption introduces systematic noise into routing supervision, making learned routing policies less reliable. To address this issue, we propose DARS (Distribution-Aware Routing Supervision), a framework that constructs routing supervision from a distributional view of model behavior. Instead of relying on a single generated response, DARS considers uncertainty from both the input side and the output side, capturing how semantically equivalent query formulations and stochastic generations affect model performance. Based on these distribution-aware observations, DARS builds more reliable supervision signals for routing. Experiments across diverse tasks show that single-shot labels can be misleading for model selection, while distribution-aware supervision provides more stable labels and improves learned routing behavior. Our results suggest that reliable LLM routing should move beyond single-response observations and be grounded in query-level model capability distributions.

The deployment of unmanned aerial vehicles (UAVs) for reliable and energy-efficient data collection from spatially distributed devices holds great promise in supporting diverse Internet of Things (IoT) applications. Nevertheless, the limited endurance and communication range of UAVs necessitate intelligent trajectory planning. While reinforcement learning (RL) has been extensively explored for UAV trajectory optimization, its interactive nature entails high costs and risks in real-world environments. Offline RL mitigates these issues but remains susceptible to unstable training and heavily rely on expert-quality datasets. To address these challenges, we formulate a joint UAV trajectory planning and resource allocation problem to maximize energy efficiency of data collection. The resource allocation subproblem is first transformed into an equivalent linear programming formulation and solved optimally with polynomial-time complexity. Then, we propose a large language model (LLM)-empowered critic-regularized decision transformer (DT) framework, termed LLM-CRDT, to learn effective UAV control policies. In LLM-CRDT, we incorporate critic networks to regularize the DT model training, thereby integrating the sequence modeling capabilities of DT with critic-based value guidance to enable learning effective policies from suboptimal datasets. Furthermore, to mitigate the data-hungry nature of transformer models, we employ a pre-trained LLM as the transformer backbone of the DT model and adopt a parameter-efficient fine-tuning strategy, i.e., LoRA, enabling rapid adaptation to UAV control tasks with small-scale dataset and low computational overhead. Extensive simulations demonstrate that LLM-CRDT outperforms benchmark online and offline RL methods, achieving up to 36.7\% higher energy efficiency than the current state-of-the-art DT approaches.

Tasks such as solving arithmetic equations, evaluating truth tables, and completing syllogisms are handled well by large language models (LLMs) in their standard form, but they often fail when the same problems are posed in logically equivalent yet obfuscated formats. To study this vulnerability, we introduce Logifus, a structure-preserving logical obfuscation framework, and, utilizing this, we present LogiQAte, a first-of-its-kind diagnostic benchmark with 1,108 questions across four reasoning tasks: (i) Obfus FOL (first-order logic entailment under equivalence-preserving rewrites), (ii) Obfus Blood Relation (family-graph entailment under indirect relational chains), (iii) Obfus Number Series (pattern induction under symbolic substitutions), and (iv) Obfus Direction Sense (navigation reasoning under altered directions and reference frames). Across all the tasks, evaluating six state-of-the-art models, we find that obfuscation severely degrades zero-shot performance, with performance dropping on average by 47% for GPT-4o, 27% for GPT-5, and 22% for reasoning model, o4-mini. Our findings reveal that current LLMs parse questions without deep understanding, highlighting the urgency of building models that genuinely comprehend and preserve meaning beyond surface form.

Transformers have achieved great success in machine learning applications. Normalization techniques, such as Layer Normalization (LayerNorm, LN) and Root Mean Square Normalization (RMSNorm), play a critical role in accelerating and stabilizing the training of Transformers. While LayerNorm recenters and rescales input vectors, RMSNorm only rescales the vectors by their RMS value. Despite being more computationally efficient, RMSNorm may compromise the representation ability of Transformers. There is currently no consensus regarding the preferred normalization technique, as some models employ LayerNorm while others utilize RMSNorm, especially in recent large language models. It is challenging to convert Transformers with one normalization to the other type. While there is an ongoing disagreement between the two normalization types, we propose a solution to unify two mainstream Transformer architectures, Pre-LN and Pre-RMSNorm Transformers. By removing the inherent redundant mean information in the main branch of Pre-LN Transformers, we can reduce LayerNorm to RMSNorm, achieving higher efficiency. We further propose the Compressed RMSNorm (CRMSNorm) and Pre-CRMSNorm Transformer based on a lossless compression of the zero-mean vectors. We formally establish the equivalence of Pre-LN, Pre-RMSNorm, and Pre-CRMSNorm Transformer variants in both training and inference. It implies that Pre-LN Transformers can be substituted with Pre-(C)RMSNorm counterparts at almost no cost, offering the same arithmetic functionality along with free efficiency improvement. Experiments demonstrate that we can reduce the training and inference time of Pre-LN Transformers by 1% - 10%.

The Empirical Interpolation Method (EIM) is a greedy procedure that constructs approximate representations of two-variable functions in separated form. In its classical presentation, the two variables play a non-symmetric role. In this work, we give an equivalent definition of the EIM approximation, in which the two variables play symmetric roles. Then, we give a proof for the existence of this approximation, and extend it up to the convergence of the EIM, and for any norm chosen to compute the error in the greedy step. Finally, we introduce a way to compute a separated representation in the case where the number of selected values is different for each variable. In the case of a physical field measured by sensors, this is useful to discard a broken sensor while keeping the information provided by the associated selected field.

Large Language Models (LLMs) are increasingly deployed in high-risk domains. However, state-of-the-art LLMs often produce hallucinations, raising serious concerns about their reliability. Prior work has explored adversarial attacks for hallucination elicitation in LLMs, but it often produces unrealistic prompts, either by inserting gibberish tokens or by altering the original meaning. As a result, these approaches offer limited insight into how hallucinations may occur in practice. While adversarial attacks in computer vision often involve realistic modifications to input images, the problem of finding realistic adversarial prompts for eliciting LLM hallucinations has remained largely underexplored. To address this gap, we propose Semantically Equivalent and Coherent Attacks (SECA) to elicit hallucinations via realistic modifications to the prompt that preserve its meaning while maintaining semantic coherence. Our contributions are threefold: (i) we formulate finding realistic attacks for hallucination elicitation as a constrained optimization problem over the input prompt space under semantic equivalence and coherence constraints; (ii) we introduce a constraint-preserving zeroth-order method to effectively search for adversarial yet feasible prompts; and (iii) we demonstrate through experiments on open-ended multiple-choice question answering tasks that SECA achieves higher attack success rates while incurring almost no semantic equivalence or semantic coherence errors compared to existing methods. SECA highlights the sensitivity of both open-source and commercial gradient-inaccessible LLMs to realistic and plausible prompt variations. Code is available at https://github.com/Buyun-Liang/SECA.

This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture, which requires the derivation of closed-form expressions for the layer-wise post-activation population covariances. In light of our results, we investigate the interplay between architecture design and implicit regularization.

We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in the paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. A direct computational implication of our results is that by solving either the barycenter problem and its dual, or the MOT problem and its dual, we can recover the optimal robust classification rule and the optimal adversarial strategy for the original adversarial problem. Examples with synthetic and real data illustrate our results.

With the growing scale and complexity of video data, efficiently processing long video sequences poses significant challenges due to the quadratic increase in memory and computational demands associated with existing transformer-based Large Multi-modal Models (LMMs). To address these issues, we introduce Video-Ma^2mba, a novel architecture that incorporates State Space Models (SSMs) within the Mamba-2 framework, replacing the attention mechanisms. This allows the LMMs to scale linearly in terms of time and memory requirements, making it feasible to handle long-duration video content. Furthermore, we enhance the memory efficiency introducing the Multi-Axis Gradient Checkpointing (MA-GC) method, which strategically manages memory by retaining only essential activations across multiple computational axes. Our approach significantly reduces the memory footprint compared to standard gradient checkpointing. Empirical analyses show that Video-Ma^2mba can process extensive video sequences-equivalent to millions of tokens or over two hours of continuous sequences at 1 FPS-on a single GPU. By maintaining a detailed capture of temporal dynamics, our model improves the accuracy and relevance of responses in long video understanding tasks, demonstrating substantial advantages over existing frameworks.

Normal-form games (NFGs) are the fundamental model of strategic interaction. We study their representation using neural networks. We describe the inherent equivariance of NFGs -- any permutation of strategies describes an equivalent game -- as well as the challenges this poses for representation learning. We then propose the NfgTransformer architecture that leverages this equivariance, leading to state-of-the-art performance in a range of game-theoretic tasks including equilibrium-solving, deviation gain estimation and ranking, with a common approach to NFG representation. We show that the resulting model is interpretable and versatile, paving the way towards deep learning systems capable of game-theoretic reasoning when interacting with humans and with each other.

Unsupervised constituency parsing focuses on identifying word sequences that form a syntactic unit (i.e., constituents) in target sentences. Linguists identify the constituent by evaluating a set of Predicate-Argument Structure (PAS) equivalent sentences where we find the constituent appears more frequently than non-constituents (i.e., the constituent corresponds to a frequent word sequence within the sentence set). However, such frequency information is unavailable in previous parsing methods that identify the constituent by observing sentences with diverse PAS. In this study, we empirically show that constituents correspond to frequent word sequences in the PAS-equivalent sentence set. We propose a frequency-based parser span-overlap that (1) computes the span-overlap score as the word sequence's frequency in the PAS-equivalent sentence set and (2) identifies the constituent structure by finding a constituent tree with the maximum span-overlap score. The parser achieves state-of-the-art level parsing accuracy, outperforming existing unsupervised parsers in eight out of ten languages. Additionally, we discover a multilingual phenomenon: participant-denoting constituents tend to have higher span-overlap scores than equal-length event-denoting constituents, meaning that the former tend to appear more frequently in the PAS-equivalent sentence set than the latter. The phenomenon indicates a statistical difference between the two constituent types, laying the foundation for future labeled unsupervised parsing research.

The Fundamental Theorem of Asset Pricing states that a market is free of arbitrage exactly when it admits an equivalent martingale measure. We formalize it in Lean 4 over Mathlib in three settings: a finite-state market over a finite horizon (Harrison-Pliska), a one-period market on an arbitrary probability space with a single scalar return (Follmer-Schied), and a one-period market with finitely many assets. The finite case is the geometry of a separating hyperplane; the scalar one-period case is an elementary change of measure. In the d-asset case the equivalent martingale measure is constructed explicitly, as the minimiser of the smooth convex potential E[log(1+e^{langleθ,Yrangle})]: absence of arbitrage is precisely coercivity of the potential, its first-order condition is the martingale property, and the minimiser's logistic weight is the density of the measure. The construction uses no Hahn-Banach theorem, no L^0-closedness argument, no measurable selection, and no non-redundancy hypothesis. To our knowledge this is the first machine-checked Fundamental Theorem of Asset Pricing in any proof assistant. The boundary is explicit: the general multi-period Dalang-Morton-Willinger theorem lies outside the development. Every theorem is sorry-free, each headline result's axioms are pinned to Mathlib's classical defaults by a build-enforced gate, and the whole is reproducible from a pinned toolchain.

A paraphrase-quality audit of MathCheck (ICLR 2025) detected 4 semantically incorrect paraphrases in 129 groups (3.1%); removing them drops GPT-4o from rank 2 to rank 4 and elevates Claude Haiku and DeepSeek V3 above it; these ranking changes are invisible to any single-model evaluation. Cross-model unanimity found these errors automatically (>= 3/4 models for MathCheck; >= 6/9 for our primary evaluation) for under $10; in our own dataset the same protocol found that 47% of auto-generated connective-variation paraphrases were semantically incorrect. That flaw compounds a deeper measurement gap: Claude Haiku 4.5 achieves 86% accuracy yet SCR=50%, meaning half its theorems are answered differently under semantically equivalent restatements, while aggregate accuracy across 9 models spans only 86-96% yet Semantic Consistency Rates (SCR) span 50-82% -- a 32-point gap invisible to standard benchmarks. Formally, for any target ranking over 9 frontier models there exists a weighting over paraphrase families that realizes it (No-Free-Benchmark corollary), because no model Pareto-dominates all families -- so benchmark designers who select families are implicitly choosing which model wins. FormInv supplies the audit protocol (replicated on external benchmarks at 100% recall), SCR and per-theorem Cochran's Q as primary invariance measures evaluated on 9 models across 366-811 items (on Lean4-verified theorems), and FormInvSelector for regime-aware model selection.

Pretraining language models on formal language can improve their acquisition of natural language. Which features of the formal language impart an inductive bias that leads to effective transfer? Drawing on insights from linguistics and complexity theory, we hypothesize that effective transfer occurs when two conditions are met: the formal language should capture the dependency structures present in natural language, and it should remain within the computational limitations of the model architecture. We experiment with pre-pretraining (training on formal language before natural languages) on transformers and find that formal languages capturing hierarchical dependencies indeed enable language models to achieve lower loss on natural language and better linguistic generalization compared to other formal languages. We also find modest support for the hypothesis that the formal language should fall within the computational limitations of the architecture. Strikingly, pre-pretraining reduces loss more efficiently than training on a matched amount of natural language. For a 1B-parameter language model trained on roughly 1.6B tokens of natural language, pre-pretraining achieves the same loss and better linguistic generalization with a 33% smaller token budget. Finally, we also give mechanistic evidence of transfer from formal to natural language: attention heads acquired during pre-pretraining remain crucial for the model's performance on syntactic evaluations.

Volatiles like H_2O are present as ice in solids in the outer cold regions of protoplanetary disks and as vapor in the warm inner regions within the water snow line. Icy pebbles drifting inwards from the outer disk sublimate after crossing the snow line, enriching the inner disk with solid mass and water vapor. Meanwhile, proto-planets forming within the disk open gaps in the disk gas, creating traps against the inward drift of pebbles and in turn reducing water enrichment in the inner disk. Recent disk observations from millimeter interferometry and infrared spectroscopy have supported this broad picture by finding a correlation between the outer radial distribution of pebbles and the properties of inner water vapor spectra. In this work, we aim at further informing previous and future observations by building on previous models to explore pebble drift in disks with multiple gaps. We systematically explore multiple gap locations and their depths (equivalent to specific masses of planets forming within), and different particle sizes to study their impact on inner disk water enrichment. We find that the presence of close-in deep gaps carved by a Jupiter-mass planet is likely crucial for blocking icy pebble delivery into the inner disk, while planets with lower masses only provide leaky traps. We also find that disks with multiple gaps show lower vapor enrichment in the inner disk. Altogether, these model results support the idea that inner disk water delivery and planet formation are regulated by the mass and location of the most massive planets.

Mathematical formulas are a fundamental and widely used component in various scientific fields, serving as a universal language for expressing complex concepts and relationships. While state-of-the-art transformer models excel in processing and understanding natural language, they encounter challenges with mathematical notation, which involves a complex structure and diverse representations. This study focuses on the development of specialized training datasets to enhance the encoding of mathematical content. We introduce Math Mutator (MAMUT), a framework capable of generating equivalent and falsified versions of a given mathematical formula in LaTeX notation, effectively capturing the mathematical variety in notation of the same concept. Based on MAMUT, we have generated four large mathematical datasets containing diverse notation, which can be used to train language models with enhanced mathematical embeddings.

Large Language Models (LLMs) for formal theorem proving have shown significant promise, yet they often lack generalizability and are fragile to even minor transformations of problem statements. To address this limitation, we introduce a novel data augmentation pipeline designed to enhance model robustness from two perspectives: symmetry and difficulty. From the symmetry perspective, we propose two complementary methods: EvolAST, an Abstract Syntax Tree (AST) based approach that targets syntactic symmetry to generate semantically equivalent problem variants, and EvolDomain, which leverages LLMs to address semantic symmetry by translating theorems across mathematical domains. From the difficulty perspective, we propose EvolDifficulty, which uses carefully designed evolutionary instructions to guide LLMs in generating new theorems with a wider range of difficulty. We then use the evolved data to train EvolProver, a 7B-parameter non-reasoning theorem prover. EvolProver establishes a new state-of-the-art (SOTA) on FormalMATH-Lite with a 53.8% pass@32 rate, surpassing all models of comparable size, including reasoning-based models. It also sets new SOTA records for non-reasoning models on MiniF2F-Test (69.8% pass@32), Ineq-Comp-Seed (52.2% pass@32), and Ineq-Comp-Transformed (34.0% pass@32). Ablation studies further confirm our data augmentation pipeline's effectiveness across multiple benchmarks.

In the complex domain of large language models (LLMs), striking a balance between computational efficiency and maintaining model quality is a formidable challenge. Navigating the inherent limitations of uniform quantization, particularly when dealing with outliers, and motivated by the launch of NVIDIA's H100 hardware, this study delves into the viability of floating-point (FP) quantization, particularly focusing on FP8 and FP4, as a potential solution. Our comprehensive investigation reveals that for LLMs, FP8 activation consistently outshines its integer (INT8) equivalent, with the performance edge becoming more noticeable in models possessing parameters beyond one billion. For weight quantization, our findings indicate that FP4 exhibits comparable, if not superior, performance to INT4, simplifying deployment on FP-supported hardware like H100. To mitigate the overhead from precision alignment caused by the disparity between weights and activations, we propose two scaling constraints for weight quantization that negligibly impact the performance compared to the standard W4A8 model. We additionally enhance our quantization methods by integrating the Low Rank Compensation (LoRC) strategy, yielding improvements especially in smaller models. The results of our investigation emphasize the immense potential of FP quantization for LLMs, paving the way for high-efficiency deployment in resource-limited settings.

Linear Transformers and State Space Models have emerged as efficient alternatives to softmax Transformers for causal sequence modeling, enabling parallel training via matrix multiplication and efficient RNN-style inference. However, despite their success in causal tasks, no unified framework exists for applying Linear Transformers to bidirectional sequence modeling. We introduce LION, the first framework to systematically extend Linear Transformers to the bidirectional setting. LION generalizes three core representations commonly used in the causal case - full Linear Attention , bidirectional RNN, and chunkwise parallel form - to the bidirectional setting. These forms are theoretically equivalent and enable models to exploit the strengths of each during training and inference. We prove that a broad class of Linear Transformers can be extended using LION and validate our framework via three core examples based on the choice of decay type: LION-LIT, the bidirectional extension of arXiv:2006.16236; LION-D, based on arXiv:2307.08621; and LION-S, a variant using selective decay arXiv:2103.02143, arXiv:2312.0075. Across standard bidirectional tasks, LION enables models to match or exceed the performance of softmax Transformers, while offering significantly faster training and more efficient inference than existing State Space Models.

The class of tree-adjoining languages can be characterized by various two-level formalisms, consisting of a context-free grammar (CFG) or pushdown automaton (PDA) controlling another CFG or PDA. These four formalisms are equivalent to tree-adjoining grammars (TAG), linear indexed grammars (LIG), pushdown-adjoining automata (PAA), and embedded pushdown automata (EPDA). We define semiring-weighted versions of the above two-level formalisms, and we design new algorithms for computing their stringsums (the weight of all derivations of a string) and allsums (the weight of all derivations). From these, we also immediately obtain stringsum and allsum algorithms for TAG, LIG, PAA, and EPDA. For LIG, our algorithm is more time-efficient by a factor of O(n|N|) (where n is the string length and |N| is the size of the nonterminal set) and more space-efficient by a factor of O(|Gamma|) (where |Gamma| is the size of the stack alphabet) than the algorithm of Vijay-Shanker and Weir (1989). For EPDA, our algorithm is both more space-efficient and time-efficient than the algorithm of Alonso et al. (2001) by factors of O(|Gamma|^2) and O(|Gamma|^3), respectively. Finally, we give the first PAA stringsum and allsum algorithms.

The growing scale of deep learning models has rendered standard hyperparameter (HP) optimization prohibitively expensive. A promising solution is the use of scale-aware hyperparameters, which can enable direct transfer of optimal HPs from small-scale grid searches to large models with minimal performance loss. To understand the principles governing such transfer strategy, we develop a general conceptual framework for reasoning about HP transfer across scale, characterizing transfer as fast when the suboptimality it induces vanishes asymptotically faster than the finite-scale performance gap. We show formally that fast transfer is equivalent to useful transfer for compute-optimal grid search, meaning that transfer is asymptotically more compute-efficient than direct tuning. While empirical work has found that the Maximal Update Parameterization (μP) exhibits fast transfer when scaling model width, the mechanisms remain poorly understood. We show that this property depends critically on problem structure by presenting synthetic settings where transfer either offers provable computational advantage or fails to outperform direct tuning even under μP. To explain the fast transfer observed in practice, we conjecture that decomposing the optimization trajectory reveals two contributions to loss reduction: (1) a width-stable component that determines the optimal HPs, and (2) a width-sensitive component that improves with width but weakly perturbs the HP optimum. We present empirical evidence for this hypothesis across various settings, including large language model pretraining.

Knowledge distillation with multiple teachers is increasingly used to improve robustness, efficiency, and safety, yet existing approaches rely largely on heuristic or implementation-specific weighting schemes. This paper develops an operator-agnostic axiomatic framework for adaptive weighting in multi-teacher knowledge distillation across three complementary scales: token, task, and context. We formalize structural conditions under which adaptive weighting operators are well-defined, admit multiple non-equivalent implementations, and can be hierarchically composed via product-structure normalization. Within this framework, we establish existence and non-uniqueness of conforming operators, characterize convergence of gradient-based optimization under standard assumptions, analyze stability and perturbation robustness, and provide an abstract formulation of safety-constrained distillation. The results decouple theoretical guarantees from specific weighting formulas, enabling principled analysis of adaptive distillation methods under heterogeneity, distribution shift, and safety constraints.

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN). We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP in terms of accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

Model attribution is a popular tool to explain the rationales behind model predictions. However, recent work suggests that the attributions are vulnerable to minute perturbations, which can be added to input samples to fool the attributions while maintaining the prediction outputs. Although empirical studies have shown positive performance via adversarial training, an effective certified defense method is eminently needed to understand the robustness of attributions. In this work, we propose to use uniform smoothing technique that augments the vanilla attributions by noises uniformly sampled from a certain space. It is proved that, for all perturbations within the attack region, the cosine similarity between uniformly smoothed attribution of perturbed sample and the unperturbed sample is guaranteed to be lower bounded. We also derive alternative formulations of the certification that is equivalent to the original one and provides the maximum size of perturbation or the minimum smoothing radius such that the attribution can not be perturbed. We evaluate the proposed method on three datasets and show that the proposed method can effectively protect the attributions from attacks, regardless of the architecture of networks, training schemes and the size of the datasets.

A homogeneous roof is a rational homogeneous variety of Picard rank 2 and index r equipped with two different mathbb P^{r-1}-bundle structures. We consider bundles of homogeneous roofs over a smooth projective variety, formulating a relative version of the duality of Calabi--Yau pairs associated to roofs of projective bundles. We discuss how derived equivalence of such pairs can lift to Calabi--Yau fibrations, extending a result of Bridgeland and Maciocia to higher-dimensional cases. We formulate an approach to prove that the DK-conjecture holds for a class of simple K-equivalent maps arising from bundles of roofs. As an example, we propose a pair of eight-dimensional Calabi--Yau varieties fibered in dual Calabi--Yau threefolds, related by a GLSM phase transition, and we prove derived equivalence with the methods above.

Tabular data in knowledge-rich domains often carries a latent prior in the form of Boolean implication relationships (BIRs) between pairs of features. We mine such relationships with a sparse-exception binomial test. The mined implications form a typed directed graph, equivalent to a propositional rule base of 2-literal clauses. We encode this graph as the connectivity of a layered neural network, called BIRDNet, in which each hidden unit corresponds to one mined rule and binds only to its two features. We show two consequences of this design: First, the architecture is sparse by construction: at most 2/d of the weights in each BIR layer are active, where d is the input dimension. Second, the model is interpretable: every trained unit keeps a stable symbolic identity, so rules can be read off the network without surrogate models. Unlike most neurosymbolic models, BIRDNet does not consume an external rule base; its structural prior is mined from the data. We evaluate BIRDNet on six transcriptomic and proteomic benchmarks. Our results show that BIRDNet stays within 0.02 AUROC of the strongest dense baseline, at a small accuracy cost, while using up to 96times fewer active parameters than an architecture-matched dense MLP. First-layer rules recover known biological signatures across multiple cancer subtypes and tissue types, including canonical amplicons, lineage-defining co-expression modules, and immune-infiltration markers. Data and code are available at: https://github.com/MAHI-Group/BIRDNet.

Large language models (LLMs) achieve strong performance across many tasks but remain vulnerable to hallucinations, making it important to systematically evaluate their reliability under realistic adversarial inputs. We formulate hallucination elicitation as a constrained optimization problem, where the goal is to find semantically coherent adversarial prompts that are equivalent to benign user prompts. Existing attack methods remain limited: discrete prompt-based attacks preserve semantic equivalence and coherence but search only over a limited set of prompt variations, while continuous latent-space attacks explore a richer space but often decode into prompts that are no longer valid rephrasings. To address these limitations, we propose REALISTA, a realistic latent-space attack framework. REALISTA constructs an input-dependent dictionary of valid editing directions, each corresponding to a semantically equivalent and coherent rephrasing, and optimizes continuous combinations of these directions in latent space. This design combines the optimization flexibility of continuous attacks with the semantic realism of discrete rephrasing-based attacks. Experiments demonstrate that REALISTA achieves superior or comparable performance to state-of-the-art realistic attacks on open-source LLMs and, crucially, succeeds in attacking large reasoning models under free-form response settings, where prior realistic attacks fail. Code is available at https://github.com/Buyun-Liang/REALISTA.

LLM agents increasingly draft messages on behalf of users, yet users routinely overshare sensitive information and disagree on what counts as private. Existing systems support only suppression (omitting sensitive information) and generalization (replacing information with an abstraction), and are typically evaluated on single isolated messages, leaving both the strategy space and evaluation setting incomplete. We formalize privacy-preserving LLM communication as an Information Sufficiency (IS) task, introduce free-text pseudonymization as a third strategy that replaces sensitive attributes with functionally equivalent alternatives, and propose a conversational evaluation protocol that assesses strategies under realistic multi-turn follow-up pressure. Across 792 scenarios spanning three power-relation types (institutional, peer, intimate) and three sensitivity categories (discrimination risk, social cost, boundary), we evaluate seven frontier LLMs on privacy at two granularities, covertness, and utility. Pseudonymization yields the strongest privacy\textendash utility tradeoff overall, and single-message evaluation systematically underestimates leakage, with generalization losing up to 16.3 percentage points of privacy under follow-up.

In this work, we study the simple yet universally applicable case of reward shaping in value-based Deep Reinforcement Learning (DRL). We show that reward shifting in the form of the linear transformation is equivalent to changing the initialization of the Q-function in function approximation. Based on such an equivalence, we bring the key insight that a positive reward shifting leads to conservative exploitation, while a negative reward shifting leads to curiosity-driven exploration. Accordingly, conservative exploitation improves offline RL value estimation, and optimistic value estimation improves exploration for online RL. We validate our insight on a range of RL tasks and show its improvement over baselines: (1) In offline RL, the conservative exploitation leads to improved performance based on off-the-shelf algorithms; (2) In online continuous control, multiple value functions with different shifting constants can be used to tackle the exploration-exploitation dilemma for better sample efficiency; (3) In discrete control tasks, a negative reward shifting yields an improvement over the curiosity-based exploration method.

Weight-space merging aims to combine multiple fine-tuned LLMs into a single model without retraining, yet most existing approaches remain fundamentally parameter-space heuristics. This creates three practical limitations. First, linear averaging, task vectors, and related rules operate on Euclidean coordinates, even though the desired goal is to merge functionality, i.e., predictive behaviors across tasks. Second, when the source checkpoints are farther apart or more heterogeneous, Euclidean blends often trigger representation collapse, manifested as activation variance shrinkage and effective-rank degradation, which sharply degrades accuracy. Third, many geometry-inspired methods are most natural for two-model interpolation and do not extend cleanly to merging N>2 experts with a principled objective. We address these issues by formulating model merging as computing a weighted Karcher mean on the Fisher--Rao manifold, which is locally equivalent to minimizing a KL-based function distance between predictive distributions. We derive a practical fixed-point algorithm using a lightweight spherical proxy that preserves norms and generalizes directly to multi-expert merging. Across various benchmarks and collapse diagnostics, our method remains stable as the number and heterogeneity of merged models increase, consistently outperforming prior baselines.

There is growing evidence of the effectiveness of Shampoo, a higher-order preconditioning method, over Adam in deep learning optimization tasks. However, Shampoo's drawbacks include additional hyperparameters and computational overhead when compared to Adam, which only updates running averages of first- and second-moment quantities. This work establishes a formal connection between Shampoo (implemented with the 1/2 power) and Adafactor -- a memory-efficient approximation of Adam -- showing that Shampoo is equivalent to running Adafactor in the eigenbasis of Shampoo's preconditioner. This insight leads to the design of a simpler and computationally efficient algorithm: ShampoO with Adam in the Preconditioner's eigenbasis (SOAP). With regards to improving Shampoo's computational efficiency, the most straightforward approach would be to simply compute Shampoo's eigendecomposition less frequently. Unfortunately, as our empirical results show, this leads to performance degradation that worsens with this frequency. SOAP mitigates this degradation by continually updating the running average of the second moment, just as Adam does, but in the current (slowly changing) coordinate basis. Furthermore, since SOAP is equivalent to running Adam in a rotated space, it introduces only one additional hyperparameter (the preconditioning frequency) compared to Adam. We empirically evaluate SOAP on language model pre-training with 360m and 660m sized models. In the large batch regime, SOAP reduces the number of iterations by over 40% and wall clock time by over 35% compared to AdamW, with approximately 20% improvements in both metrics compared to Shampoo. An implementation of SOAP is available at https://github.com/nikhilvyas/SOAP.

While prompt engineering is instrumental in maximizing the capabilities of Large Language Models (LLMs) during inference, the role of prompts during training remains critically underexplored. Prevailing fine-tuning paradigms typically treat training prompts as mere surface forms, assuming that semantically equivalent instructions yield identical learning outcomes. However, we reveal that this equivalence is deceptive: while paraphrased prompts often lead to comparable in-task performance, they induce drastically different cross-task impacts regarding catastrophic forgetting and generalization. Crucially, these impacts are positively correlated across tasks, indicating the existence of superior prompts that consistently yield better performance. Furthermore, we discover that these superior prompts can be robustly identified by task loss prior to learning. Leveraging these insights, we introduce State-Adaptive Prompt Optimization (SAPO), a lightweight yet effective training strategy that shifts task formulation from a static input to a dynamic, state-adaptive variable. Comprehensive experiments on diverse benchmarks confirm its effectiveness, which significantly mitigates forgetting while improving generalization, achieving substantial performance gains over state-of-the-art methods. These results provide insights into how training prompts shape learning dynamics and offer a practical recipe for robust fine-tuning. Our code is available at https://github.com/Eric8932/SAPO.

Deep residual architectures, such as ResNet and the Transformer, have enabled models of unprecedented depth, yet a formal understanding of why depth is so effective remains an open question. A popular intuition, following Veit et al. (2016), is that these residual networks behave like ensembles of many shallower models. Our key finding is an explicit analytical formula that verifies this ensemble perspective, proving that increasing network depth is mathematically equivalent to expanding the size of this implicit ensemble. Furthermore, our expansion reveals a hierarchical ensemble structure in which the combinatorial growth of computation paths leads to an explosion in the output signal, explaining the historical necessity of normalization layers in training deep models. This insight offers a first principles explanation for the historical dependence on normalization layers and sheds new light on a family of successful normalization-free techniques like SkipInit and Fixup. However, while these previous approaches infer scaling factors through optimizer analysis or a heuristic analogy to Batch Normalization, our work offers the first explanation derived directly from the network's inherent functional structure. Specifically, our Residual Expansion Theorem reveals that scaling each residual module provides a principled solution to taming the combinatorial explosion inherent to these architectures. We further show that this scaling acts as a capacity controls that also implicitly regularizes the model's complexity.

A natural strategy for continual learning is to weigh a Bayesian ensemble of fixed functions. This suggests that if a (single) neural network could be interpreted as an ensemble, one could design effective algorithms that learn without forgetting. To realize this possibility, we observe that a neural network classifier with N parameters can be interpreted as a weighted ensemble of N classifiers, and that in the lazy regime limit these classifiers are fixed throughout learning. We call these classifiers the neural tangent experts and show they output valid probability distributions over the labels. We then derive the likelihood and posterior probability of each expert given past data. Surprisingly, the posterior updates for these experts are equivalent to a scaled and projected form of stochastic gradient descent (SGD) over the network weights. Away from the lazy regime, networks can be seen as ensembles of adaptive experts which improve over time. These results offer a new interpretation of neural networks as Bayesian ensembles of experts, providing a principled framework for understanding and mitigating catastrophic forgetting in continual learning settings.

Olympiad-level benchmarks in mathematics and physics are crucial testbeds for advanced AI reasoning, but chemistry, with its unique multimodal symbolic language, has remained an open challenge. We introduce ChemO, a new benchmark built from the International Chemistry Olympiad (IChO) 2025. ChemO features two key innovations for automated assessment: Assessment-Equivalent Reformulation (AER), which converts problems requiring visual outputs (e.g., drawing molecules) into computationally tractable formats, and Structured Visual Enhancement (SVE), a diagnostic mechanism to disentangle a model's visual perception capabilities from its core chemical reasoning. To tackle this benchmark, we propose ChemLabs, a hierarchical multi-agent framework that mimics human expert collaboration through specialized agents for problem decomposition, perception, reasoning, and auditing. Experiments on state-of-the-art multimodal models demonstrate that combining SVE with our multi-agent system yields dramatic performance gains. Our top configuration achieves a score of 93.6 out of 100, surpassing an estimated human gold medal threshold and establishing a new state-of-the-art in automated chemical problem-solving. ChemO Dataset: https://huggingface.co/datasets/IDEA-AI4SCI/ChemO

While semi-supervised learning (SSL) has yielded promising results, the more realistic SSL scenario remains to be explored, in which the unlabeled data exhibits extremely high recognition difficulty, e.g., fine-grained visual classification in the context of SSL (SS-FGVC). The increased recognition difficulty on fine-grained unlabeled data spells disaster for pseudo-labeling accuracy, resulting in poor performance of the SSL model. To tackle this challenge, we propose Soft Label Selection with Confidence-Aware Clustering based on Class Transition Tracking (SoC) by reconstructing the pseudo-label selection process by jointly optimizing Expansion Objective and Shrinkage Objective, which is based on a soft label manner. Respectively, the former objective encourages soft labels to absorb more candidate classes to ensure the attendance of ground-truth class, while the latter encourages soft labels to reject more noisy classes, which is theoretically proved to be equivalent to entropy minimization. In comparisons with various state-of-the-art methods, our approach demonstrates its superior performance in SS-FGVC. Checkpoints and source code are available at https://github.com/NJUyued/SoC4SS-FGVC.

Floor labels of crowdsourced RF signals are crucial for many smart-city applications, such as multi-floor indoor localization, geofencing, and robot surveillance. To build a prediction model to identify the floor number of a new RF signal upon its measurement, conventional approaches using the crowdsourced RF signals assume that at least few labeled signal samples are available on each floor. In this work, we push the envelope further and demonstrate that it is technically feasible to enable such floor identification with only one floor-labeled signal sample on the bottom floor while having the rest of signal samples unlabeled. We propose FIS-ONE, a novel floor identification system with only one labeled sample. FIS-ONE consists of two steps, namely signal clustering and cluster indexing. We first build a bipartite graph to model the RF signal samples and obtain a latent representation of each node (each signal sample) using our attention-based graph neural network model so that the RF signal samples can be clustered more accurately. Then, we tackle the problem of indexing the clusters with proper floor labels, by leveraging the observation that signals from an access point can be detected on different floors, i.e., signal spillover. Specifically, we formulate a cluster indexing problem as a combinatorial optimization problem and show that it is equivalent to solving a traveling salesman problem, whose (near-)optimal solution can be found efficiently. We have implemented FIS-ONE and validated its effectiveness on the Microsoft dataset and in three large shopping malls. Our results show that FIS-ONE outperforms other baseline algorithms significantly, with up to 23% improvement in adjusted rand index and 25% improvement in normalized mutual information using only one floor-labeled signal sample.

The attention module is the key component in Transformers. While the global attention mechanism offers high expressiveness, its excessive computational cost restricts its applicability in various scenarios. In this paper, we propose a novel attention paradigm, Agent Attention, to strike a favorable balance between computational efficiency and representation power. Specifically, the Agent Attention, denoted as a quadruple (Q, A, K, V), introduces an additional set of agent tokens A into the conventional attention module. The agent tokens first act as the agent for the query tokens Q to aggregate information from K and V, and then broadcast the information back to Q. Given the number of agent tokens can be designed to be much smaller than the number of query tokens, the agent attention is significantly more efficient than the widely adopted Softmax attention, while preserving global context modelling capability. Interestingly, we show that the proposed agent attention is equivalent to a generalized form of linear attention. Therefore, agent attention seamlessly integrates the powerful Softmax attention and the highly efficient linear attention. Extensive experiments demonstrate the effectiveness of agent attention with various vision Transformers and across diverse vision tasks, including image classification, object detection, semantic segmentation and image generation. Notably, agent attention has shown remarkable performance in high-resolution scenarios, owning to its linear attention nature. For instance, when applied to Stable Diffusion, our agent attention accelerates generation and substantially enhances image generation quality without any additional training. Code is available at https://github.com/LeapLabTHU/Agent-Attention.

Pre-trained large language model (LLM) is under exploration to perform NLP tasks that may require logical reasoning. Logic-driven data augmentation for representation learning has been shown to improve the performance of tasks requiring logical reasoning, but most of these data rely on designed templates and therefore lack generalization. In this regard, we propose an AMR-based logical equivalence-driven data augmentation method (AMR-LE) for generating logically equivalent data. Specifically, we first parse a text into the form of an AMR graph, next apply four logical equivalence laws (contraposition, double negation, commutative and implication laws) on the AMR graph to construct a logically equivalent/inequivalent AMR graph, and then convert it into a logically equivalent/inequivalent sentence. To help the model to better learn these logical equivalence laws, we propose a logical equivalence-driven contrastive learning training paradigm, which aims to distinguish the difference between logical equivalence and inequivalence. Our AMR-LE (Ensemble) achieves #2 on the ReClor leaderboard https://eval.ai/web/challenges/challenge-page/503/leaderboard/1347 . Our model shows better performance on seven downstream tasks, including ReClor, LogiQA, MNLI, MRPC, RTE, QNLI, and QQP. The source code and dataset are public at https://github.com/Strong-AI-Lab/Logical-Equivalence-driven-AMR-Data-Augmentation-for-Representation-Learning .

We introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver. Our system demonstrates state-of-the-art performance with favorable scaling properties for automated theorem proving.

We present NeRF-XL, a principled method for distributing Neural Radiance Fields (NeRFs) across multiple GPUs, thus enabling the training and rendering of NeRFs with an arbitrarily large capacity. We begin by revisiting existing multi-GPU approaches, which decompose large scenes into multiple independently trained NeRFs, and identify several fundamental issues with these methods that hinder improvements in reconstruction quality as additional computational resources (GPUs) are used in training. NeRF-XL remedies these issues and enables the training and rendering of NeRFs with an arbitrary number of parameters by simply using more hardware. At the core of our method lies a novel distributed training and rendering formulation, which is mathematically equivalent to the classic single-GPU case and minimizes communication between GPUs. By unlocking NeRFs with arbitrarily large parameter counts, our approach is the first to reveal multi-GPU scaling laws for NeRFs, showing improvements in reconstruction quality with larger parameter counts and speed improvements with more GPUs. We demonstrate the effectiveness of NeRF-XL on a wide variety of datasets, including the largest open-source dataset to date, MatrixCity, containing 258K images covering a 25km^2 city area.

Reinforcement Learning From Human Feedback (RLHF) has been a critical to the success of the latest generation of generative AI models. In response to the complex nature of the classical RLHF pipeline, direct alignment algorithms such as Direct Preference Optimization (DPO) have emerged as an alternative approach. Although DPO solves the same objective as the standard RLHF setup, there is a mismatch between the two approaches. Standard RLHF deploys reinforcement learning in a specific token-level MDP, while DPO is derived as a bandit problem in which the whole response of the model is treated as a single arm. In this work we rectify this difference, first we theoretically show that we can derive DPO in the token-level MDP as a general inverse Q-learning algorithm, which satisfies the Bellman equation. Using our theoretical results, we provide three concrete empirical insights. First, we show that because of its token level interpretation, DPO is able to perform some type of credit assignment. Next, we prove that under the token level formulation, classical search-based algorithms, such as MCTS, which have recently been applied to the language generation space, are equivalent to likelihood-based search on a DPO policy. Empirically we show that a simple beam search yields meaningful improvement over the base DPO policy. Finally, we show how the choice of reference policy causes implicit rewards to decline during training. We conclude by discussing applications of our work, including information elicitation in multi-tun dialogue, reasoning, agentic applications and end-to-end training of multi-model systems.

Structured State-Space Duality (SSD) [Dao & Gu, ICML 2024] is an equivalence between a simple Structured State-Space Model (SSM) and a masked attention mechanism. In particular, a state-space model with a scalar-times-identity state matrix is equivalent to a masked self-attention with a 1-semiseparable causal mask. Consequently, the same sequence transformation (model) has two algorithmic realizations: as a linear-time O(T) recurrence or as a quadratic-time O(T^2) attention. In this note, we formalize and generalize this duality: (i) we extend SSD from the scalar-identity case to general diagonal SSMs (diagonal state matrices); (ii) we show that these diagonal SSMs match the scalar case's training complexity lower bounds while supporting richer dynamics; (iii) we establish a necessary and sufficient condition under which an SSM is equivalent to 1-semiseparable masked attention; and (iv) we show that such duality fails to extend to standard softmax attention due to rank explosion. Together, these results tighten bridge between recurrent SSMs and Transformers, and widen the design space for expressive yet efficient sequence models.

Blueshifted absorption is the classic spectroscopic signature of an accretion disc wind in X-ray binaries and cataclysmic variables (CVs). However, outflows can also create pure emission lines, especially at optical wavelengths. Therefore, developing other outflow diagnostics for these types of lines is worthwhile. With this in mind, we construct a systematic grid of 3645 synthetic wind-formed H-alpha line profiles for CVs with the radiative transfer code SIROCCO. Our grid yields a variety of line shapes: symmetric, asymmetric, single- to quadruple-peaked, and even P-Cygni profiles. About 20% of these lines -- our `Gold' sample -- have strengths and widths consistent with observations. We use this grid to test a recently proposed method for identifying wind-formed emission lines based on deviations in the wing profile shape: the `excess equivalent width diagnostic diagram'. We find that our `Gold' sample can preferentially populate the suggested `wind regions' of this diagram. However, the method is highly sensitive to the adopted definition of the line profile `wing'. Hence, we propose a refined definition based on the full-width at half maximum to improve the interpretability of the diagnostic diagram. Furthermore, we define an approximate scaling relation for the strengths of wind-formed CV emission lines in terms of the outflow parameters. This relation provides a fast way to assess whether -- and what kind of -- outflow can produce an observed emission line. All our wind-based models are open-source and we provide an easy-to-use web-based tool to browse our full set of H-alpha spectral profiles.

Implementing Boolean functions with circuits consisting of logic gates is fundamental in digital computer design. However, the implemented circuit must be exactly equivalent, which hinders generative neural approaches on this task due to their occasionally wrong predictions. In this study, we introduce a generative neural model, the "Circuit Transformer", which eliminates such wrong predictions and produces logic circuits strictly equivalent to given Boolean functions. The main idea is a carefully designed decoding mechanism that builds a circuit step-by-step by generating tokens, which has beneficial "cutoff properties" that block a candidate token once it invalidate equivalence. In such a way, the proposed model works similar to typical LLMs while logical equivalence is strictly preserved. A Markov decision process formulation is also proposed for optimizing certain objectives of circuits. Experimentally, we trained an 88-million-parameter Circuit Transformer to generate equivalent yet more compact forms of input circuits, outperforming existing neural approaches on both synthetic and real world benchmarks, without any violation of equivalence constraints.

Large language models (LLMs) have revolutionized artificial intelligence, yet their massive memory and computational demands necessitate aggressive quantization, increasingly pushing representations toward the theoretical limit of a single bit. While complex-valued LLMs, such as iFairy, offer a superior chance for low-bit representation compared to real-valued counterparts, they require training from scratch, preventing the utilization of the vast ecosystem of pre-trained real-valued foundation models. Here we present Fairy2i, a universal framework that transforms pre-trained real-valued layers into an equivalent widely-linear complex form, enabling extremely low-bit quantization while reusing existing checkpoints. By proving a lossless mathematical equivalence between real and widely-linear maps, we convert standard Transformers into the complex domain and employ a phase-aware quantization scheme with a highly efficient codebook of fourth roots of unity. Furthermore, we introduce a recursive residual quantization mechanism that iteratively minimizes quantization error, allowing inference to proceed via efficient multiplication-free accumulation. We demonstrate that Fairy2i restores the performance of LLaMA-2 7B at an effective 2-bit precision to levels nearly comparable with full-precision baselines, significantly outperforming state-of-the-art real-valued binary and ternary quantization methods. This work bridges the gap between the representational efficiency of complex-valued arithmetic and the practical utility of pre-trained models, paving a new way for efficient inference on commodity hardware.

Score-based diffusion models generate samples from an unknown target distribution using a time-reversed diffusion process. While such models represent state-of-the-art approaches in industrial applications such as artificial image generation, it has recently been noted that their performance can be further improved by considering injection noise with heavy tailed characteristics. Here, I present a generalization of generative diffusion processes to a wide class of non-Gaussian noise processes. I consider forward processes driven by standard Gaussian noise with super-imposed Poisson jumps representing a finite activity Levy process. The generative process is shown to be governed by a generalized score function that depends on the jump amplitude distribution and can be estimated by minimizing a simple MSE loss as in conventional Gaussian models. Both probability flow ODE and SDE formulations are derived using basic technical effort. A detailed implementation for a pure jump process with Laplace distributed amplitudes yields a generalized score function in closed analytical form and is shown to outperform the equivalent Gaussian model in specific parameter regimes.

Recent advances in prompt learning allow large language model agents to acquire task-relevant knowledge from inference-time context without parameter changes. For example, existing methods (like ACE or GEPA) can learn system prompts to improve accuracy based on previous agent runs. However, these methods primarily focus on single-agent or low-parallelism settings. This fundamentally limits their ability to efficiently learn from a large set of collected agentic traces. It would be efficient and beneficial to run prompt learning in parallel to accommodate the growing trend of learning from many agentic traces or parallel agent executions. Yet without a principled strategy for scaling, current methods suffer from quality degradation with high parallelism. To improve both the efficiency and quality of prompt learning, we propose Combee, a novel framework to scale parallel prompt learning for self-improving agents. Combee speeds up learning and enables running many agents in parallel while learning from their aggregate traces without quality degradation. To achieve this, Combee leverages parallel scans and employs an augmented shuffle mechanism; Combee also introduces a dynamic batch size controller to balance quality and delay. Evaluations on AppWorld, Terminal-Bench, Formula, and FiNER demonstrate that Combee achieves up to 17x speedup over previous methods with comparable or better accuracy and equivalent cost.

Large Language Models (LLMs) have emerged as a milestone in artificial intelligence, and their performance can improve as the model size increases. However, this scaling brings great challenges to training and inference efficiency, particularly for deploying LLMs in resource-constrained environments, and the scaling trend is becoming increasingly unsustainable. This paper introduces the concept of ``capacity density'' as a new metric to evaluate the quality of the LLMs across different scales and describes the trend of LLMs in terms of both effectiveness and efficiency. To calculate the capacity density of a given target LLM, we first introduce a set of reference models and develop a scaling law to predict the downstream performance of these reference models based on their parameter sizes. We then define the effective parameter size of the target LLM as the parameter size required by a reference model to achieve equivalent performance, and formalize the capacity density as the ratio of the effective parameter size to the actual parameter size of the target LLM. Capacity density provides a unified framework for assessing both model effectiveness and efficiency. Our further analysis of recent open-source base LLMs reveals an empirical law (the densing law)that the capacity density of LLMs grows exponentially over time. More specifically, using some widely used benchmarks for evaluation, the capacity density of LLMs doubles approximately every three months. The law provides new perspectives to guide future LLM development, emphasizing the importance of improving capacity density to achieve optimal results with minimal computational overhead.

As large language models (LLMs) are increasingly applied to scientific reasoning, the complexity of answer formats and the diversity of equivalent expressions make answer verification a critical yet challenging task. Existing verification studies in scientific domains suffer from two major limitations: (a) the absence of systematic evaluation standards and insufficient disciplinary coverage, which hinders their comprehensive assessment; and (b) heavy reliance on cumbersome rule design or prompt engineering, which reduces their effectiveness in complex reasoning scenarios or limits their cross-disciplinary generalization. To address these challenges, we propose solutions at both the data and model levels. On the data side, we construct SCI-VerifyBench, a cross-disciplinary benchmark covering mathematics, physics, biology, chemistry, and general scientific QA. The benchmark is built from real LLM responses and enhanced with domain-specific equivalence transformations that generate challenging and realistic data. Model-based and expert annotations ensure both quality and diversity, enabling rigorous evaluation of verification ability. On the model side, we emphasize the importance of reasoning for verification and introduce SCI-Verifier, a unified reasoning-augmented verifier for scientific domains. Through post-training, SCI-Verifier demonstrates strong logical reasoning and equivalence judgment capabilities while maintaining concise and stable outputs. Together, SCI-VerifyBench and SCI-Verifier provide a principled framework for scientific verification, offering both systematic evaluation and practical pathways to enhance the reliability and applicability of LLMs in scientific domains.

Trustworthy verifiers are essential for the success of reinforcement learning with verifiable reward (RLVR), which is the core methodology behind various large reasoning models such as DeepSeek-R1. In complex domains like mathematical reasoning, rule-based verifiers have been widely adopted in previous works to train strong reasoning models. However, the reliability of these verifiers and their impact on the RL training process remain poorly understood. In this work, we take mathematical reasoning as a case study and conduct a comprehensive analysis of various verifiers in both static evaluation and RL training scenarios. First, we find that current open-source rule-based verifiers often fail to recognize equivalent answers presented in different formats across multiple commonly used mathematical datasets, resulting in non-negligible false negative rates. This limitation adversely affects RL training performance and becomes more pronounced as the policy model gets stronger. Subsequently, we investigate model-based verifiers as a potential solution to address these limitations. While the static evaluation shows that model-based verifiers achieve significantly higher verification accuracy, further analysis and RL training results imply that they are highly susceptible to hacking, where they misclassify certain patterns in responses as correct (i.e., false positives). This vulnerability is exploited during policy model optimization, leading to artificially inflated rewards. Our findings underscore the unique risks inherent to both rule-based and model-based verifiers, aiming to offer valuable insights to develop more robust reward systems in reinforcement learning.

Continuous representations of logic formulae allow us to integrate symbolic knowledge into data-driven learning algorithms. If such embeddings are semantically consistent, i.e. if similar specifications are mapped into nearby vectors, they enable continuous learning and optimization directly in the semantic space of formulae. However, to translate the optimal continuous representation into a concrete requirement, such embeddings must be invertible. We tackle this issue by training a Transformer-based decoder-only model to invert semantic embeddings of Signal Temporal Logic (STL) formulae. STL is a powerful formalism that allows us to describe properties of signals varying over time in an expressive yet concise way. By constructing a small vocabulary from STL syntax, we demonstrate that our proposed model is able to generate valid formulae after only 1 epoch and to generalize to the semantics of the logic in about 10 epochs. Additionally, the model is able to decode a given embedding into formulae that are often simpler in terms of length and nesting while remaining semantically close (or equivalent) to gold references. We show the effectiveness of our methodology across various levels of training formulae complexity to assess the impact of training data on the model's ability to effectively capture the semantic information contained in the embeddings and generalize out-of-distribution. Finally, we deploy our model for solving a requirement mining task, i.e. inferring STL specifications that solve a classification task on trajectories, performing the optimization directly in the semantic space.

Communication compression, a technique aiming to reduce the information volume to be transmitted over the air, has gained great interests in Federated Learning (FL) for the potential of alleviating its communication overhead. However, communication compression brings forth new challenges in FL due to the interplay of compression-incurred information distortion and inherent characteristics of FL such as partial participation and data heterogeneity. Despite the recent development, the performance of compressed FL approaches has not been fully exploited. The existing approaches either cannot accommodate arbitrary data heterogeneity or partial participation, or require stringent conditions on compression. In this paper, we revisit the seminal stochastic controlled averaging method by proposing an equivalent but more efficient/simplified formulation with halved uplink communication costs. Building upon this implementation, we propose two compressed FL algorithms, SCALLION and SCAFCOM, to support unbiased and biased compression, respectively. Both the proposed methods outperform the existing compressed FL methods in terms of communication and computation complexities. Moreover, SCALLION and SCAFCOM accommodates arbitrary data heterogeneity and do not make any additional assumptions on compression errors. Experiments show that SCALLION and SCAFCOM can match the performance of corresponding full-precision FL approaches with substantially reduced uplink communication, and outperform recent compressed FL methods under the same communication budget.

Improvements in training data scale and quality have led to significant advances, yet its influence in speech recognition remains underexplored. In this paper, we present a large-scale dataset, OLMoASR-Pool, and series of models, OLMoASR, to study and develop robust zero-shot speech recognition models. Beginning from OLMoASR-Pool, a collection of 3M hours of English audio and 17M transcripts, we design text heuristic filters to remove low-quality or mistranscribed data. Our curation pipeline produces a new dataset containing 1M hours of high-quality audio-transcript pairs, which we call OLMoASR-Mix. We use OLMoASR-Mix to train the OLMoASR-Mix suite of models, ranging from 39M (tiny.en) to 1.5B (large.en) parameters. Across all model scales, OLMoASR achieves comparable average performance to OpenAI's Whisper on short and long-form speech recognition benchmarks. Notably, OLMoASR-medium.en attains a 12.8\% and 11.0\% word error rate (WER) that is on par with Whisper's largest English-only model Whisper-medium.en's 12.4\% and 10.5\% WER for short and long-form recognition respectively (at equivalent parameter count). OLMoASR-Pool, OLMoASR models, and filtering, training and evaluation code will be made publicly available to further research on robust speech processing.

Each LoRA checkpoint compactly stores task-specific updates in low-rank weight matrices, offering an efficient way to adapt large language models to new tasks and domains. In principle, these weights already encode what the adapter does and how well it performs. In this paper, we ask whether this information can be read directly from the weights, without running the base model or accessing training data. A key obstacle is that a single LoRA update can be factorized in infinitely many ways. Without resolving this ambiguity, models trained on the factors may fit the particular factorization rather than the underlying update. To this end, we propose \methodfull, which maps each LoRA update to a provably canonical form via QR decomposition followed by SVD, so that all equivalent factorizations share the same representation. The resulting components are then tokenized and processed by a Transformer to produce a weight-space embedding. Across language and vision LoRA collections, W2T achieves strong results on attribute classification, performance prediction, and adapter retrieval, demonstrating that LoRA weights reliably indicate model behavior once factorization ambiguity is removed. Code is available at https://github.com/xiaolonghan2000/Weight2Token.

Reservoir Computing (RC) has established itself as an efficient paradigm for temporal processing. However, its scalability remains severely constrained by (i) the necessity of processing temporal data sequentially and (ii) the prohibitive memory footprint of high-dimensional reservoirs. In this work, we revisit RC through the lens of structured operators and state space modeling to address these limitations, introducing Parallel Echo State Network (ParalESN). ParalESN enables the construction of high-dimensional and efficient reservoirs based on diagonal linear recurrence in the complex space, enabling parallel processing of temporal data. We provide a theoretical analysis demonstrating that ParalESN preserves the Echo State Property and the universality guarantees of traditional Echo State Networks while admitting an equivalent representation of arbitrary linear reservoirs in the complex diagonal form. Empirically, ParalESN matches the predictive accuracy of traditional RC on time series benchmarks, while delivering substantial computational savings. On 1-D pixel-level classification tasks, ParalESN achieves competitive accuracy with fully trainable neural networks while reducing computational costs and energy consumption by orders of magnitude. Overall, ParalESN offers a promising, scalable, and principled pathway for integrating RC within the deep learning landscape.

Idioms pose a fundamental challenge for language models, as their meaning cannot be inferred from surface form alone. Understanding such expressions, therefore, requires semantic abstraction beyond lexical overlap. We introduce IdioLink, a retrieval benchmark designed to test whether models can link idiomatic expressions to conceptually equivalent meanings expressed in literal or paraphrased forms. IdioLink comprises 10,700 documents and 2,140 queries, spanning 107 idioms with both literal and figurative uses. Each document and query is annotated with spans that convey the core meaning. Evaluating strong embedding baselines (e.g., BGE, E5, Contriever, and Qwen), we show that current models struggle to retrieve equivalent meanings across divergent surface realizations, relying instead on topical and shallow semantic cues. IdioLink exposes key gaps in idiom-aware semantic retrieval and provides a challenging testbed for future models.

Modern text-to-image diffusion models encode rich visual priors, but expose them only through one-way text-conditioned generation. Existing unified vision--language models derived from them recover bidirectional capability through large-scale joint pretraining or substantial retraining of the text pathway, discarding the strong image prior the text-to-image backbone already encodes. We introduce FullFlow, a parameter-efficient recipe that upgrades a pretrained rectified-flow text-to-image model into a bidirectional vision--language generator by training only LoRA adapters and lightweight text heads. FullFlow keeps images in their native continuous flow and adds a discrete insertion process for text. Separate image and text timesteps turn inference into trajectory selection in a two-dimensional generative space, enabling textrightarrowimage, imagerightarrowtext, joint sampling, and partial-text prediction with a single backbone. On Stable Diffusion 3 (SD3) under an identical trainable-parameter count and matched LoRA rank, FullFlow improves textrightarrowimage FID from 62.7 to 31.6 and imagerightarrowtext CIDEr from 2.0 to 99.4 over a LoRA equivalent following the previous SOTA formulation (Dual Diffusion) at matched wall-clock training time, while reducing peak VRAM from {sim}84\,GB to {sim}38\,GB and raising throughput by {sim}8times on two RTX A5000 GPUs in under 24 hours, training only {sim}5% of the backbone parameters. The same recipe transfers to FLUX.1-dev and supports downstream VQA through partial-text generation. These results show that strong bidirectional vision--language capability can be unlocked from pretrained text-to-image flow models without full multimodal pretraining.

Sequence learning reduces to similarity-based retrieval over a temporally indexed representation space, a constraint on any sequence model, not a property of a specific architecture. We show that a spiking Sparse Distributed Memory sequence machine (2007) and the transformer (2017) independently instantiate the same five functional operations (encoding, context maintenance, associative retrieval, storage, and decoding), with cosine similarity as the shared retrieval primitive in both. We formalise a Phase-Latency Isomorphism showing that sinusoidal positional phase and spike timing are linearly related, and prove that dot product attention is invariant to this mapping up to a global scale factor on the positional component (Lemma 1). Empirically, frequency-compressed positional encoding fails to converge on a positionally demanding copy task, while a learned rank-based embedding matches or exceeds sinusoidal encoding, indicating that the critical property for positional representation is distance discriminability under dot-product similarity, not sinusoidal form. Time, phase, and rank are three instantiations of the same computational primitive, an ordered index whose structure survives similarity-based retrieval.

Recent work on large language models has demonstrated the use of model-free reinforcement learning (RL) to train reasoning-like capabilities. The emergence of "thinking" through model-free RL is interesting as thinking actions neither produce reward nor change the external world state to one where the agent is more likely to get reward. This paper seeks to build a domain-independent understanding of when model-free RL will lead to such "thinking" as a strategy for reward maximization. To build this understanding, we first introduce a theoretical model which we call a thought Markov decision process (MDP). Thought MDPs minimally extend the classical MDP model to include an abstract notion of thought state and thought action. Using the thought MDP model, we prove the importance of policy initialization in determining whether or not thinking emerges and show formally that thought actions are equivalent to the agent choosing to perform a step of policy improvement before continuing to act. We then show that open-source LLMs satisfy the conditions that our theory predicts are necessary for model-free RL to produce thinking-like behavior. Finally, we hypothesize sufficient conditions that would enable thinking to be learned outside of language generation and introduce a toy domain where a combination of multi-task pre-training and designated thought actions enable more data-efficient RL compared to non-thinking agents.

While measuring bias and robustness in coreference resolution are important goals, such measurements are only as good as the tools we use to measure them. Winogender Schemas (Rudinger et al., 2018) are an influential dataset proposed to evaluate gender bias in coreference resolution, but a closer look reveals issues with the data that compromise its use for reliable evaluation, including treating different pronominal forms as equivalent, violations of template constraints, and typographical errors. We identify these issues and fix them, contributing a new dataset: WinoPron. Using WinoPron, we evaluate two state-of-the-art supervised coreference resolution systems, SpanBERT, and five sizes of FLAN-T5, and demonstrate that accusative pronouns are harder to resolve for all models. We also propose a new method to evaluate pronominal bias in coreference resolution that goes beyond the binary. With this method, we also show that bias characteristics vary not just across pronoun sets (e.g., he vs. she), but also across surface forms of those sets (e.g., him vs. his).

Speech Large Language Models (Speech LLMs) lack a principled mechanism for streaming inference: their label-synchronous generation has no acoustic-frame alignment, making real-time decoding and end-of-utterance detection difficult. We propose TRADE TRansducer-Augmented DEcoder, which augments a multimodal LLM with a transducer branch that shares the audio encoder and uses the LLM's hidden states directly as the prediction network -- coupling frame-synchronous acoustic alignment with the LLM's linguistic reasoning. Three design choices make the system accurate, streamable, and long-form capable: (1)Tightly coupled dual vocabularies -- a compact transducer vocabulary derived from the LLM vocabulary, enabling zero-cost score fusion; (2)Chunk-synchronized streaming training with gradient stopping, eliminating the train-inference mismatch at offline-equivalent memory cost; and (3)Localized Decoder Audio Attention (LDAA), a causal sliding window that caps KV-cache memory independently of utterance length. A single TRADE checkpoint supports offline and streaming decoding across a continuous range of latency operating points. TRADE achieves 6.71% average WER on the Open ASR Leaderboard, while the streaming recognition with 960ms chunk size reaches 8.40% from the same checkpoint. On long-form speech, it obtains 3.64% WER on TED-LIUM and 10.88% on Earnings-22 without external segmentation. TRADE provides sentence-end punctuation timestamps that, when combined with acoustic voice activity detection (VAD), improve end-of-utterance detection by +0.03 F_1 over acoustic VAD alone.

Flow matching models (FMs) have revolutionized text-to-image (T2I) generation, with reinforcement learning (RL) serving as a critical post-training strategy for alignment with reward objectives. In this research, we show that current RL pipelines for FMs suffer from two underappreciated yet important limitations: sample inefficiency due to insufficient generation diversity, and pronounced prompt overfitting, where models memorize specific training formulations and exhibit dramatic performance collapse when evaluated on semantically equivalent but stylistically varied prompts. We present PromptRL (Prompt Matters in RL for Flow-Based Image Generation), a framework that incorporates language models (LMs) as trainable prompt refinement agents directly within the flow-based RL optimization loop. This design yields two complementary benefits: rapid development of sophisticated prompt rewriting capabilities and, critically, a synergistic training regime that reshapes the optimization dynamics. PromptRL achieves state-of-the-art performance across multiple benchmarks, obtaining scores of 0.97 on GenEval, 0.98 on OCR accuracy, and 24.05 on PickScore. Furthermore, we validate the effectiveness of our RL approach on large-scale image editing models, improving the EditReward of FLUX.1-Kontext from 1.19 to 1.43 with only 0.06 million rollouts, surpassing Gemini 2.5 Flash Image (also known as Nano Banana), which scores 1.37, and achieving comparable performance with ReasonNet (1.44), which relied on fine-grained data annotations along with a complex multi-stage training. Our extensive experiments empirically demonstrate that PromptRL consistently achieves higher performance ceilings while requiring over 2times fewer rollouts compared to naive flow-only RL. Our code is available at https://github.com/G-U-N/UniRL.

Classifier-Free Guidance (CFG) has recently emerged in text-to-image generation as a lightweight technique to encourage prompt-adherence in generations. In this work, we demonstrate that CFG can be used broadly as an inference-time technique in pure language modeling. We show that CFG (1) improves the performance of Pythia, GPT-2 and LLaMA-family models across an array of tasks: Q\&A, reasoning, code generation, and machine translation, achieving SOTA on LAMBADA with LLaMA-7B over PaLM-540B; (2) brings improvements equivalent to a model with twice the parameter-count; (3) can stack alongside other inference-time methods like Chain-of-Thought and Self-Consistency, yielding further improvements in difficult tasks; (4) can be used to increase the faithfulness and coherence of assistants in challenging form-driven and content-driven prompts: in a human evaluation we show a 75\% preference for GPT4All using CFG over baseline.

Cross-tokenizer knowledge distillation allows a student model to learn from teachers with incompatible vocabularies. Prior work operates on hidden states or logits; the latter is preferred as a drop-in replacement requiring no auxiliary components. Logit-based methods either use only the correct-token probability, missing the full 'dark knowledge' in the teacher's distribution, or operate on the full output distribution, relying on strict token partitioning and/or unprincipled heuristic ranking. We identify two key shortcomings of full-distribution, logit-based methods: (i) an uncommon-token failure, where critical tokens fall into the unmatched subset (e.g., Llama's 1100 multi-digit numerals under digit-splitting Qwen supervision) and are suppressed during training, reducing GSM8k from 12.89 to 2.56 compared to same-tokenizer KD from a weaker teacher; and (ii) over-conservative matching, where strict 1-to-1 matching excludes near-equivalent tokens across surface forms. These failures require distinct remedies: eliminating the partition when critical tokens are misaligned, and refining it when alignment is reliable. We propose X-Token, an approach with two complementary loss formulations targeting these issues. P-KL removes partitioning and aligns the student's distribution with the teacher's via a sparse projection matrix W (initialized from tokenizer-level string rules) to address the uncommon-token failure. H-KL retains the hybrid form while relaxing matching to align each student token with its top-ranked teacher mapping under W. Both objectives share W and extend naturally to multiple teachers. Empirically, on Llama-3.2-1B, X-Token outperforms the current state of the art GOLD by +3.82 average points with a Qwen3-4B teacher and by +0.5 with a Phi-4-Mini teacher. Further, a two-teacher setup (Phi-4-mini + Llama-3B) improves over single-teacher distillation by +1.3 points.

In this paper, we study the ideas of composition and decomposition in the context of vector spaces, graphs and matroids. For vector spaces V_{AB}, treated as collection of row vectors, with specified column set Auplus B, we define V_{SP}lrarv V_{PQ}, Scap Q= emptyset, to be the collection of all vectors (f_S,f_Q) such that (f_S,f_P)in V_{SP}, (f_P,f_Q)in V_{PQ}. An analogous operation G_{SP}lrarg G_{PQ}equivd G_{PQ} can be defined in relation to graphs G_{SP}, G_{PQ}, on edge sets Suplus P, Puplus Q, respectively in terms of an overlapping subgraph G_P which gets deleted in the right side graph (see for instance the notion of k-sum oxley). For matroids we define the `linking' M_{SP}lrarm M_{PQ} equivd (M_{SP}vee M_{PQ})times (Suplus Q), denoting the contraction operation by 'times'. In each case, we examine how to minimize the size of the `overlap' set P, without affecting the right side entity. In the case of vector spaces, there is a polynomial time algorithm for achieving the minimum, which we present. Similar ideas work for graphs and for matroids under appropriate conditions. Next we consider the problem of decomposition. Here, in the case of vector spaces, the problem is to decompose V_{SQ} as V_{SP}lrarv V_{PQ}, with minimum size P. We give a polynomial time algorithm for this purpose. In the case of graphs and matroids we give a solution to this problem under certain restrictions.

A formal system called cologic is proposed for the study of compacta. A counterpart of countable model theory is developed for this system, and it is applied to model theory of the pseudo-arc.

As models and data scale, independently trained networks often induce analogous notions of similarity. But, matching similarities is weaker than establishing an explicit correspondence between the representation spaces, especially for multimodal models, where consistency must hold not only within each modality, but also for the learned image-text coupling. We therefore ask: given two independently trained multimodal contrastive models (with encoders (f, g) and (f,g)) -- trained on different distributions and with different architectures -- does a systematic geometric relationship exist between their embedding spaces? If so, what form does it take, and does it hold uniformly across modalities? In this work, we show that across model families such as CLIP, SigLIP, and FLAVA, this geometric relationship is well approximated by an orthogonal map (up to a global mean shift), i.e., there exists an orthogonal map Q where Q^top Q = I such that f(x)approx Q f(x) for paired images x. Strikingly, the same Q simultaneously aligns the text encoders i.e., g(y)approx Q g(y) for texts y. Theoretically, we prove that if the multimodal kernel agrees across models on a small anchor set i.e. langle f(x), g(y)rangle approx langle f(x), g(y)rangle, then the two models must be related by a single orthogonal map Q and the same Q maps images and text across models. More broadly, this finding enables backward-compatible model upgrades, avoiding costly re-embedding, and has implications for the privacy of learned representations. Our project page: https://canonical-multimodal.github.io/

We propose a novel application of prompting Pre-trained Language Models (PLMs) to generate analogies and study how to design effective prompts for two task settings: generating a source concept analogous to a given target concept (aka Analogous Concept Generation or ACG), and generating an explanation of the similarity between a given pair of target concept and source concept (aka Analogous Explanation Generation or AEG). We found that it is feasible to prompt InstructGPT to generate meaningful analogies and the best prompts tend to be precise imperative statements especially with a low temperature setting. We also systematically analyzed the sensitivity of the InstructGPT model to prompt design, temperature, and injected spelling errors, and found that the model is particularly sensitive to certain variations (e.g., questions vs. imperative statements). Further, we conducted human evaluation on 1.4k of the generated analogies and found that the quality of generations varies substantially by model size. The largest InstructGPT model can achieve human-level performance at generating meaningful analogies for a given target while there is still room for improvement on the AEG task.

We present a smoothly broken power law functional form (that we refer to as a Broken Neural Scaling Law (BNSL)) that accurately models & extrapolates the scaling behaviors of deep neural networks (i.e. how the evaluation metric of interest varies as amount of compute used for training (or inference), number of model parameters, training dataset size, model input size, number of training steps, or upstream performance varies) for various architectures & for each of various tasks within a large & diverse set of upstream & downstream tasks, in zero-shot, prompted, & finetuned settings. This set includes large-scale vision, language, audio, video, diffusion, generative modeling, multimodal learning, contrastive learning, AI alignment, AI capabilities, robotics, out-of-distribution (OOD) generalization, continual learning, transfer learning, uncertainty estimation / calibration, OOD detection, adversarial robustness, distillation, sparsity, retrieval, quantization, pruning, fairness, molecules, computer programming/coding, math word problems, "emergent phase transitions", arithmetic, supervised learning, unsupervised/self-supervised learning, & reinforcement learning (single agent & multi-agent). When compared to other functional forms for neural scaling, this functional form yields extrapolations of scaling behavior that are considerably more accurate on this set. Moreover, this functional form accurately models & extrapolates scaling behavior that other functional forms are incapable of expressing such as the nonmonotonic transitions present in the scaling behavior of phenomena such as double descent & the delayed, sharp inflection points present in the scaling behavior of tasks such as arithmetic. Lastly, we use this functional form to glean insights about the limit of the predictability of scaling behavior. Code is available at https://github.com/ethancaballero/broken_neural_scaling_laws

Large Language Models (LLMs) excel at complex reasoning through search algorithms, yet current strategies often suffer from massive token consumption due to redundant exploration of semantically equivalent steps. Existing semantic similarity methods struggle to accurately identify such equivalence in domain-specific contexts like mathematical reasoning. To address this, we propose EquivPruner, a simple yet effective approach that identifies and prunes semantically equivalent actions during LLM reasoning search. We also introduce MathEquiv, the first dataset we created for mathematical statement equivalence, which enables the training of a lightweight equivalence detector. Extensive experiments across various models and tasks demonstrate that EquivPruner significantly reduces token consumption, improving searching efficiency and often bolstering reasoning accuracy. For instance, when applied to Qwen2.5-Math-7B-Instruct on GSM8K, EquivPruner reduced token consumption by 48.1\% while also improving accuracy. Our code is available at https://github.com/Lolo1222/EquivPruner.

Symbolic regression seeks to uncover physical laws from experimental data by searching for closed-form expressions, which is an important task in AI-driven scientific discovery. Yet the exponential growth of the search space of expression renders the task computationally challenging. A promising yet underexplored direction for reducing the search space and accelerating training lies in *symbolic equivalence*: many expressions, although syntactically different, define the same function -- for example, log(x_1^2x_2^3), log(x_1^2)+log(x_2^3), and 2log(x_1)+3log(x_2). Existing algorithms treat such variants as distinct outputs, leading to redundant exploration and slow learning. We introduce EGG-SR, a unified framework that integrates symbolic equivalence into a class of modern symbolic regression methods, including Monte Carlo Tree Search (MCTS), Deep Reinforcement Learning (DRL), and Large Language Models (LLMs). EGG-SR compactly represents equivalent expressions through the proposed EGG module (via equality graphs), accelerating learning by: (1) pruning redundant subtree exploration in EGG-MCTS, (2) aggregating rewards across equivalent generated sequences in EGG-DRL, and (3) enriching feedback prompts in EGG-LLM. Theoretically, we show the benefit of embedding EGG into learning: it tightens the regret bound of MCTS and reduces the variance of the DRL gradient estimator. Empirically, EGG-SR consistently enhances a class of symbolic regression models across several benchmarks, discovering more accurate expressions within the same time limit. Project page is at: https://nan-jiang-group.github.io/egg-sr.

Given appropriate representations of the semantic relations between carpenter and wood and between mason and stone (for example, vectors in a vector space model), a suitable algorithm should be able to recognize that these relations are highly similar (carpenter is to wood as mason is to stone; the relations are analogous). Likewise, with representations of dog, house, and kennel, an algorithm should be able to recognize that the semantic composition of dog and house, dog house, is highly similar to kennel (dog house and kennel are synonymous). It seems that these two tasks, recognizing relations and compositions, are closely connected. However, up to now, the best models for relations are significantly different from the best models for compositions. In this paper, we introduce a dual-space model that unifies these two tasks. This model matches the performance of the best previous models for relations and compositions. The dual-space model consists of a space for measuring domain similarity and a space for measuring function similarity. Carpenter and wood share the same domain, the domain of carpentry. Mason and stone share the same domain, the domain of masonry. Carpenter and mason share the same function, the function of artisans. Wood and stone share the same function, the function of materials. In the composition dog house, kennel has some domain overlap with both dog and house (the domains of pets and buildings). The function of kennel is similar to the function of house (the function of shelters). By combining domain and function similarities in various ways, we can model relations, compositions, and other aspects of semantics.

Under ZF, we show that the statement that every subset of every R-vector space has a maximal convex subset is equivalent to the Axiom of Choice. We also study the strength of the same statement restricted to some specific R-vector spaces. In particular, we show that the statement for R^2 is equivalent to the Axiom of Countable Choice for reals, whereas the statement for R^3 is equivalent to the Axiom of Uniformization. We discuss the statement for some spaces of higher dimensions as well.

The behavior of neural networks still remains opaque, and a recently widely noted phenomenon is that networks often achieve similar performance when initialized with different random parameters. This phenomenon has attracted significant attention in measuring the similarity between features learned by distinct networks. However, feature similarity could be vague in describing the same feature since equivalent features hardly exist. In this paper, we expand the concept of equivalent feature and provide the definition of what we call functionally equivalent features. These features produce equivalent output under certain transformations. Using this definition, we aim to derive a more intrinsic metric for the so-called feature complexity regarding the redundancy of features learned by a neural network at each layer. We offer a formal interpretation of our approach through the lens of category theory, a well-developed area in mathematics. To quantify the feature complexity, we further propose an efficient algorithm named Iterative Feature Merging. Our experimental results validate our ideas and theories from various perspectives. We empirically demonstrate that the functionally equivalence widely exists among different features learned by the same neural network and we could reduce the number of parameters of the network without affecting the performance.The IFM shows great potential as a data-agnostic model prune method. We have also drawn several interesting empirical findings regarding the defined feature complexity.

We use new interior models of cold planets to investigate the mass-radius relationships of solid exoplanets, considering planets made primarily of iron, silicates, water, and carbon compounds. We find that the mass-radius relationships for cold terrestrial-mass planets of all compositions we considered follow a generic functional form that is not a simple power law: log_{10} R_s = k_1 + 1/3 log_{10}(M_s) - k_2 M_s^{k_3} for up to M_p approx 20 M_{oplus}, where M_s and R_s are scaled mass and radius values. This functional form arises because the common building blocks of solid planets all have equations of state that are well approximated by a modified polytrope of the form rho = rho_0 + c P^n. We find that highly detailed planet interior models, including temperature structure and phase changes, are not necessary to derive solid exoplanet bulk composition from mass and radius measurements. For solid exoplanets with no substantial atmosphere we have also found that: with 5% fractional uncertainty in planet mass and radius it is possible to distinguish among planets composed predominantly of iron or silicates or water ice but not more detailed compositions; with sim~5% uncertainty water ice planets with gtrsim 25% water by mass may be identified; the minimum plausible planet size for a given mass is that of a pure iron planet; and carbon planet mass-radius relationships overlap with those of silicate and water planets due to similar zero-pressure densities and equations of state. We propose a definition of "super Earths'' based on the clear distinction in radii between planets with significant gas envelopes and those without.

English. This document aims to study the best algorithms to verify the belonging of a specific document to a related domain by comparing different methods for calculating the distance between two vectors. This study has been made possible with the help of the structures made available by the Apache Spark framework. Starting from the study illustrated in the publication "New frontier of textual classification: Big data and distributed calculus" by Massimiliano Morrelli et al., We wanted to carry out a study on the possible implementation of a solution capable of calculating the Similarity of a sentence using the distributed environment. Italiano. Il presente documento persegue l'obiettivo di studiare gli algoritmi migliori per verificare l'appartenenza di un determinato documento a un relativo dominio tramite un confronto di diversi metodi per il calcolo della distanza fra due vettori. Tale studio \`e stato condotto con l'ausilio delle strutture messe a disposizione dal framework Apache Spark. Partendo dallo studio illustrato nella pubblicazione "Nuova frontiera della classificazione testuale: Big data e calcolo distribuito" di Massimiliano Morrelli et al., si \`e voluto realizzare uno studio sulla possibile implementazione di una soluzione in grado di calcolare la Similarit\`a di una frase sfruttando l'ambiente distribuito.

We present a complete formalization in Isabelle/HOL of the object part of an equivalence between L-mosaics and bounded join-semilattices, employing an AI-assisted methodology that integrates large language models as reasoning assistants throughout the proof development process. The equivalence was originally established by Cangiotti, Linzi, and Talotti in their study of hypercompositional structures related to orthomodular lattices and quantum logic. Our formalization rigorously verifies the main theoretical result and demonstrates the mutual inverse property of the transformations establishing this equivalence. The development showcases both the mathematical depth of multivalued algebraic operations and the potential for AI-enhanced interactive theorem proving in tackling complex formalization projects.

Zero-shot transfer learning for document understanding is a crucial yet under-investigated scenario to help reduce the high cost involved in annotating document entities. We present a novel query-based framework, QueryForm, that extracts entity values from form-like documents in a zero-shot fashion. QueryForm contains a dual prompting mechanism that composes both the document schema and a specific entity type into a query, which is used to prompt a Transformer model to perform a single entity extraction task. Furthermore, we propose to leverage large-scale query-entity pairs generated from form-like webpages with weak HTML annotations to pre-train QueryForm. By unifying pre-training and fine-tuning into the same query-based framework, QueryForm enables models to learn from structured documents containing various entities and layouts, leading to better generalization to target document types without the need for target-specific training data. QueryForm sets new state-of-the-art average F1 score on both the XFUND (+4.6%~10.1%) and the Payment (+3.2%~9.5%) zero-shot benchmark, with a smaller model size and no additional image input.

We propose a novel approach for learning interchangeable tokens in language models to obtain an extendable vocabulary that can generalize to new tokens. Our method is designed to address alpha-equivalence, the principle that renaming bound variables in a syntactic expression preserves semantics. This property arises in many formal languages such as temporal logics, in which all proposition symbols represent the same concept but are distinguishable from each other. To handle such tokens, we develop a dual-part embedding approach. The first part is shared across all interchangeable tokens, thereby enforcing that they represent the same core concept. The second part is randomly generated for each token, which enables distinguishability. We evaluate our method in a Transformer encoder-decoder model on two tasks: solving linear temporal logic formulae and copying with extendable vocabulary. Our method demonstrates promising generalization capabilities in addition to introducing a favorable inductive bias for alpha-equivalence.

Recent research shows that the dynamics of an infinitely wide neural network (NN) trained by gradient descent can be characterized by Neural Tangent Kernel (NTK) jacot2018neural. Under the squared loss, the infinite-width NN trained by gradient descent with an infinitely small learning rate is equivalent to kernel regression with NTK arora2019exact. However, the equivalence is only known for ridge regression currently arora2019harnessing, while the equivalence between NN and other kernel machines (KMs), e.g. support vector machine (SVM), remains unknown. Therefore, in this work, we propose to establish the equivalence between NN and SVM, and specifically, the infinitely wide NN trained by soft margin loss and the standard soft margin SVM with NTK trained by subgradient descent. Our main theoretical results include establishing the equivalences between NNs and a broad family of ell_2 regularized KMs with finite-width bounds, which cannot be handled by prior work, and showing that every finite-width NN trained by such regularized loss functions is approximately a KM. Furthermore, we demonstrate our theory can enable three practical applications, including (i) non-vacuous generalization bound of NN via the corresponding KM; (ii) non-trivial robustness certificate for the infinite-width NN (while existing robustness verification methods would provide vacuous bounds); (iii) intrinsically more robust infinite-width NNs than those from previous kernel regression. Our code for the experiments is available at https://github.com/leslie-CH/equiv-nn-svm.

Inverse scattering problem for an operator, which is a sum of the operator of the third derivative and of an operator of multiplication by a real function, is solved. The main closed system of equations of inverse problem is obtained. This system contains scattering coefficients and bound state elements as independent parameters. Form of the simplest reflectionless potential (analogous to soliton) is found.

Let G be a graph with adjacency matrix A(G) and degree matrix D(G), and let L_μ(G):=A(G)-μD(G). Two graphs G_1 and G_2 are called degree-similar if there exists an invertible matrix M such that M^{-1} A(G_1) M =A(G_2) and M^{-1} D(G_1) M =D(G_2). In this paper, we address three problems concerning degree-similar graphs proposed by Godsil and Sun. First, we present a new characterization of degree-similar graphs using degree partition, from which we derive methods and examples for constructing cospectral graphs and degree-similar graphs. Second, we construct infinite pairs of non-degree-similar trees G_1 and G_2 such that tI- L_μ(G_1) and tI-L_μ(G_2) have the same Smith normal form over Q(μ)[t], which provides a negative answer to a problem posed by Godsil and Sun. Third, we establish several invariants of degree-similar graphs and obtain results on unicyclic graphs that are degree-similar determined. Lastly we prove that for a strongly regular graph G and any two edges e and f of G, G backslash e and G backslash f have identical μ-polynomial, i.e., det(tI-L_μ(G backslash e))=det(tI-L_μ(G backslash f)), which enables the construction of pairs of non-isomorphic graphs with same μ-polynomial, where G backslash e denotes the graph obtained from G by deleting the edge e.

Since long, research on machine translation has been ongoing. Still, we do not get good translations from MT engines so developed. Manual ranking of these outputs tends to be very time consuming and expensive. Identifying which one is better or worse than the others is a very taxing task. In this paper, we show an approach which can provide automatic ranks to MT outputs (translations) taken from different MT Engines and which is based on N-gram approximations. We provide a solution where no human intervention is required for ranking systems. Further we also show the evaluations of our results which show equivalent results as that of human ranking.

The predictions of question answering (QA)systems are typically evaluated against manually annotated finite sets of one or more answers. This leads to a coverage limitation that results in underestimating the true performance of systems, and is typically addressed by extending over exact match (EM) with pre-defined rules or with the token-level F1 measure. In this paper, we present the first systematic conceptual and data-driven analysis to examine the shortcomings of token-level equivalence measures. To this end, we define the asymmetric notion of answer equivalence (AE), accepting answers that are equivalent to or improve over the reference, and publish over 23k human judgments for candidates produced by multiple QA systems on SQuAD. Through a careful analysis of this data, we reveal and quantify several concrete limitations of the F1 measure, such as a false impression of graduality, or missing dependence on the question. Since collecting AE annotations for each evaluated model is expensive, we learn a BERT matching (BEM) measure to approximate this task. Being a simpler task than QA, we find BEM to provide significantly better AE approximations than F1, and to more accurately reflect the performance of systems. Finally, we demonstrate the practical utility of AE and BEM on the concrete application of minimal accurate prediction sets, reducing the number of required answers by up to x2.6.

Binding precedents (S\'umulas Vinculantes) constitute a juridical instrument unique to the Brazilian legal system and whose objectives include the protection of the Federal Supreme Court against repetitive demands. Studies of the effectiveness of these instruments in decreasing the Court's exposure to similar cases, however, indicate that they tend to fail in such a direction, with some of the binding precedents seemingly creating new demands. We empirically assess the legal impact of five binding precedents, 11, 14, 17, 26 and 37, at the highest court level through their effects on the legal subjects they address. This analysis is only possible through the comparison of the Court's ruling about the precedents' themes before they are created, which means that these decisions should be detected through techniques of Similar Case Retrieval. The contributions of this article are therefore twofold: on the mathematical side, we compare the uses of different methods of Natural Language Processing -- TF-IDF, LSTM, BERT, and regex -- for Similar Case Retrieval, whereas on the legal side, we contrast the inefficiency of these binding precedents with a set of hypotheses that may justify their repeated usage. We observe that the deep learning models performed significantly worse in the specific Similar Case Retrieval task and that the reasons for binding precedents to fail in responding to repetitive demand are heterogeneous and case-dependent, making it impossible to single out a specific cause.

We introduce Legal RAG Bench, a benchmark and evaluation methodology for assessing the end-to-end performance of legal RAG systems. As a benchmark, Legal RAG Bench consists of 4,876 passages from the Victorian Criminal Charge Book alongside 100 complex, hand-crafted questions demanding expert knowledge of criminal law and procedure. Both long-form answers and supporting passages are provided. As an evaluation methodology, Legal RAG Bench leverages a full factorial design and novel hierarchical error decomposition framework, enabling apples-to-apples comparisons of the contributions of retrieval and reasoning models in RAG. We evaluate three state-of-the-art embedding models (Isaacus' Kanon 2 Embedder, Google's Gemini Embedding 001, and OpenAI's Text Embedding 3 Large) and two frontier LLMs (Gemini 3.1 Pro and GPT-5.2), finding that information retrieval is the primary driver of legal RAG performance, with LLMs exerting a more moderate effect on correctness and groundedness. Kanon 2 Embedder, in particular, had the largest positive impact on performance, improving average correctness by 17.5 points, groundedness by 4.5 points, and retrieval accuracy by 34 points. We observe that many errors attributed to hallucinations in legal RAG systems are in fact triggered by retrieval failures, concluding that retrieval sets the ceiling for the performance of many modern legal RAG systems. We document why and how we built Legal RAG Bench alongside the results of our evaluations. We also openly release our code and data to assist with reproduction of our findings.

Making analogies is fundamental to cognition. Proportional analogies, which consist of four terms, are often used to assess linguistic and cognitive abilities. For instance, completing analogies like "Oxygen is to Gas as <blank> is to <blank>" requires identifying the semantic relationship (e.g., "type of") between the first pair of terms ("Oxygen" and "Gas") and finding a second pair that shares the same relationship (e.g., "Aluminum" and "Metal"). In this work, we introduce a 15K Multiple-Choice Question Answering (MCQA) dataset for proportional analogy completion and evaluate the performance of contemporary Large Language Models (LLMs) in various knowledge-enhanced prompt settings. Specifically, we augment prompts with three types of knowledge: exemplar, structured, and targeted. Our results show that despite extensive training data, solving proportional analogies remains challenging for current LLMs, with the best model achieving an accuracy of 55%. Notably, we find that providing targeted knowledge can better assist models in completing proportional analogies compared to providing exemplars or collections of structured knowledge.

We present a functional form (that we refer to as a Unified Neural Scaling Law (UNSL)) that accurately models and extrapolates the scaling behaviors of deep neural networks as multiple dimensions all vary simultaneously (i.e. how the evaluation metric of interest varies as one simultaneously varies the number of model parameters, training dataset size, number of training steps, number of inference steps, amount of compute, and various hyperparameters) for various architectures and for each of various tasks within a varied set of upstream and downstream tasks. This set includes large-scale vision, language, math, and reinforcement learning. When compared to other functional forms for neural scaling, this functional form yields extrapolations of scaling behavior that are considerably more accurate on this set.

Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and non-existence pieces of C.

By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that there are an infinite number of equivalence classes of Gauss words under homotopy.

Embedding based product recommendations have gained popularity in recent years due to its ability to easily integrate to large-scale systems and allowing nearest neighbor searches in real-time. The bulk of studies in this area has predominantly been focused on similar item recommendations. Research on complementary item recommendations, on the other hand, still remains considerably under-explored. We define similar items as items that are interchangeable in terms of their utility and complementary items as items that serve different purposes, yet are compatible when used with one another. In this paper, we apply a novel approach to finding complementary items by leveraging dual embedding representations for products. We demonstrate that the notion of relatedness discovered in NLP for skip-gram negative sampling (SGNS) models translates effectively to the concept of complementarity when training item representations using co-purchase data. Since sparsity of purchase data is a major challenge in real-world scenarios, we further augment the model using synthetic samples to extend coverage. This allows the model to provide complementary recommendations for items that do not share co-purchase data by leveraging other abundantly available data modalities such as images, text, clicks etc. We establish the effectiveness of our approach in improving both coverage and quality of recommendations on real world data for a major online retail company. We further show the importance of task specific hyperparameter tuning in training SGNS. Our model is effective yet simple to implement, making it a great candidate for generating complementary item recommendations at any e-commerce website.

We construct a gauged linear sigma model with two non-birational K\"alher phases which we prove to be derived equivalent, L-equivalent, deformation equivalent and Hodge equivalent. This provides a new counterexample to the birational Torelli problem which admits a simple GLSM interpretation.

Modern retrieval system often requires recomputing the representation of every piece of data in the gallery when updating to a better representation model. This process is known as backfilling and can be especially costly in the real world where the gallery often contains billions of samples. Recently, researchers have proposed the idea of Backward Compatible Training (BCT) where the new representation model can be trained with an auxiliary loss to make it backward compatible with the old representation. In this way, the new representation can be directly compared with the old representation, in principle avoiding the need for any backfilling. However, followup work shows that there is an inherent tradeoff where a backward compatible representation model cannot simultaneously maintain the performance of the new model itself. This paper reports our ``not-so-surprising'' finding that adding extra dimensions to the representation can help here. However, we also found that naively increasing the dimension of the representation did not work. To deal with this, we propose Backward-compatible Training with a novel Basis Transformation (BT^2). A basis transformation (BT) is basically a learnable set of parameters that applies an orthonormal transformation. Such a transformation possesses an important property whereby the original information contained in its input is retained in its output. We show in this paper how a BT can be utilized to add only the necessary amount of additional dimensions. We empirically verify the advantage of BT^2 over other state-of-the-art methods in a wide range of settings. We then further extend BT^2 to other challenging yet more practical settings, including significant change in model architecture (CNN to Transformers), modality change, and even a series of updates in the model architecture mimicking the evolution of deep learning models.

Current unlearning metrics for generative models evaluate success based on reference responses or classifier outputs rather than assessing the core objective: whether the unlearned model behaves indistinguishably from a model that never saw the unwanted data. This reference-specific approach creates systematic blind spots, allowing models to appear successful while retaining unwanted knowledge accessible through alternative prompts or attacks. We address these limitations by proposing Functional Alignment for Distributional Equivalence (FADE), a novel metric that measures distributional similarity between unlearned and reference models by comparing bidirectional likelihood assignments over generated samples. Unlike existing approaches that rely on predetermined references, FADE captures functional alignment across the entire output distribution, providing a principled assessment of genuine unlearning. Our experiments on the TOFU benchmark for LLM unlearning and the UnlearnCanvas benchmark for text-to-image diffusion model unlearning reveal that methods achieving near-optimal scores on traditional metrics fail to achieve distributional equivalence, with many becoming more distant from the gold standard than before unlearning. These findings expose fundamental gaps in current evaluation practices and demonstrate that FADE provides a more robust foundation for developing and assessing truly effective unlearning methods.

By introducing a dynamical version of the second fundamental form, we generalize a recent result of Filip-Fisher-Lowe to the setting of magnetic systems. Namely, we show that a real-analytic negatively s-curved magnetic system on a closed real-analytic manifold has only finitely many closed totally s-magnetic hypersurfaces, unless the magnetic 2-form is trivial and the underlying metric is hyperbolic.

Strategies for the generation of periodic discrete structures with identical two-point correlation are developed. Starting from a pair of root structures, which are not related by translation, phase inversion or axis reflections, child structures of arbitrary resolution (i.e., pixel or voxel numbers) and number of phases (i.e., material phases/species) can be generated by means of trivial embedding based phase extension, application of kernels and/or phase coalescence, such that the generated structures inherit the two-point-correlation equivalence. Proofs of the inheritance property are provided by means of the Discrete Fourier Transform theory. A Python 3 implementation of the results is offered by the authors through the Github repository https://github.com/DataAnalyticsEngineering/EQ2PC in order to make the provided results reproducible and useful for all interested readers. Examples for the generation of structures are demonstrated, together with applications in the homogenization theory of periodic media.

We train a bilingual Arabic-Hebrew language model using a transliterated version of Arabic texts in Hebrew, to ensure both languages are represented in the same script. Given the morphological, structural similarities, and the extensive number of cognates shared among Arabic and Hebrew, we assess the performance of a language model that employs a unified script for both languages, on machine translation which requires cross-lingual knowledge. The results are promising: our model outperforms a contrasting model which keeps the Arabic texts in the Arabic script, demonstrating the efficacy of the transliteration step. Despite being trained on a dataset approximately 60% smaller than that of other existing language models, our model appears to deliver comparable performance in machine translation across both translation directions.

When researchers ask whether two transformer layers are "equivalent" for compression, they often conflate distinct tests. Replacement asks whether one layer's map can substitute for another's in place; interchange asks whether two layers approximately commute when their positions are swapped. Both are output-grounded swap-KL probes, but they need not agree: on pretrained transformers the protocol gap can change which layers look safe to prune by several-fold under the same evaluator, especially when replacement distances are high. We measure both protocols across checkpoints and architectures. On a Pythia training trajectory (410M and 1.4B), the replacement-interchange gap grows from initialization to convergence. Under one matched WikiText-2 contract at 8B scale, Qwen3-8B enters a divergent regime: interchange-guided removal is several-fold safer than replacement-guided at the same layer budgets, while Llama-3.1-8B ties the two protocols for pruning cost even though interchange KL is lower, showing metric gaps need not map one-to-one to removal. Before layer removal or merging, score both swap-KLs on the target checkpoint; the diagnostic requires only unlabeled forward passes.

This document offers a detailed linguistic description of SNACS (Semantic Network of Adposition and Case Supersenses; Schneider et al., 2018), an inventory of 52 semantic labels ("supersenses") that characterize the use of adpositions and case markers at a somewhat coarse level of granularity, as demonstrated in the STREUSLE corpus (https://github.com/nert-nlp/streusle/ ; version 4.5 tracks guidelines version 2.6). Though the SNACS inventory aspires to be universal, this document is specific to English; documentation for other languages will be published separately. Version 2 is a revision of the supersense inventory proposed for English by Schneider et al. (2015, 2016) (henceforth "v1"), which in turn was based on previous schemes. The present inventory was developed after extensive review of the v1 corpus annotations for English, plus previously unanalyzed genitive case possessives (Blodgett and Schneider, 2018), as well as consideration of adposition and case phenomena in Hebrew, Hindi, Korean, and German. Hwang et al. (2017) present the theoretical underpinnings of the v2 scheme. Schneider et al. (2018) summarize the scheme, its application to English corpus data, and an automatic disambiguation task. Liu et al. (2021) offer an English Lexical Semantic Recognition tagger that includes SNACS labels in its output. This documentation can also be browsed alongside corpus data on the Xposition website (Gessler et al., 2022): http://www.xposition.org/

We present a system that uses LLMs as a tool in the development of Constructed Languages. The system is modular in that one first creates a target phonology for the language using an agentic approach that refines its output at each step with commentary feedback on its previous attempt. Next, a set of sentences is 'translated' from their English original into a morphosyntactic markup that reflects the word order and morphosyntactic feature specifications of the desired target language, with affixes represented as morphosyntactic feature bundles. From this translated corpus, a lexicon is constructed using the phonological model and the set of morphemes (stems and affixes) extracted from the 'translated' sentences. The system is then instructed to provide an orthography for the language, using an existing script such as Latin or Cyrillic. Finally, the system writes a brief grammatical handbook of the language. The system can also translate further sentences into the target language. Our goal is twofold. First, we hope that these tools will be fun to use for creating artificially constructed languages. Second, we are interested in exploring what LLMs 'know' about language-not what they know about any particular language or linguistic phenomenon, but how much they know about and understand language and linguistic concepts. As we shall see, there is a fairly wide gulf in capabilities both among different LLMs and among different linguistic specifications, with it being notably easier for systems to deal with more common patterns than rarer ones. An additional avenue that we explore is the application of our approach to translating from high-resource into low-resource languages. While the results so far are mostly negative, we provide some evidence that an improved version of the present system could afford some real gains in such tasks. https://github.com/SakanaAI/IASC

The procure to pay process (P2P) in large enterprises is a back-end business process which deals with the procurement of products and services for enterprise operations. Procurement is done by issuing purchase orders to impaneled vendors and invoices submitted by vendors are paid after they go through a rigorous validation process. Agents orchestrating P2P process often encounter the problem of matching a product or service descriptions in the invoice to those in purchase order and verify if the ordered items are what have been supplied or serviced. For example, the description in the invoice and purchase order could be TRES 739mL CD KER Smooth and TRES 0.739L CD KER Smth which look different at word level but refer to the same item. In a typical P2P process, agents are asked to manually select the products which are similar before invoices are posted for payment. This step in the business process is manual, repetitive, cumbersome, and costly. Since descriptions are not well-formed sentences, we cannot apply existing semantic and syntactic text similarity approaches directly. In this paper, we present two approaches to solve the above problem using various types of available agent's recorded feedback data. If the agent's feedback is in the form of a relative ranking between descriptions, we use similarity ranking algorithm. If the agent's feedback is absolute such as match or no-match, we use classification similarity algorithm. We also present the threats to the validity of our approach and present a possible remedy making use of product taxonomy and catalog. We showcase the comparative effectiveness and efficiency of the proposed approaches over many benchmarks and real-world data sets.

We prove that: I. If L is a T_1 space, |L|>1 and d(L) leq kappa geq omega, then there is a submaximal dense subspace X of L^{2^kappa} such that |X|=Delta(X)=kappa; II. If cleqkappa=kappa^omega<lambda and 2^kappa=2^lambda, then there is a Tychonoff pseudocompact globally and locally connected space X such that |X|=Delta(X)=lambda and X is not kappa^+-resolvable; III. If omega_1leqkappa<lambda and 2^kappa=2^lambda, then there is a regular space X such that |X|=Delta(X)=lambda, all continuous real-valued functions on X are constant (so X is pseudocompact and connected) and X is not kappa^+-resolvable.

The paper focuses on deep learning semantic search algorithms applied in the HR domain. The aim of the article is developing a novel approach to training a Siamese network to link the skills mentioned in the job ad with the title. It has been shown that the title normalization process can be based either on classification or similarity comparison approaches. While classification algorithms strive to classify a sample into predefined set of categories, similarity search algorithms take a more flexible approach, since they are designed to find samples that are similar to a given query sample, without requiring pre-defined classes and labels. In this article semantic similarity search to find candidates for title normalization has been used. A pre-trained language model has been adapted while teaching it to match titles and skills based on co-occurrence information. For the purpose of this research fifty billion title-descriptions pairs had been collected for training the model and thirty three thousand title-description-normalized title triplets, where normalized job title was picked up manually by job ad creator for testing purposes. As baselines FastText, BERT, SentenceBert and JobBert have been used. As a metric of the accuracy of the designed algorithm is Recall in top one, five and ten model's suggestions. It has been shown that the novel training objective lets it achieve significant improvement in comparison to other generic and specific text encoders. Two settings with treating titles as standalone strings, and with included skills as additional features during inference have been used and the results have been compared in this article. Improvements by 10% and 21.5% have been achieved using VacancySBERT and VacancySBERT (with skills) respectively. The benchmark has been developed as open-source to foster further research in the area.

When robots learn reward functions using high capacity models that take raw state directly as input, they need to both learn a representation for what matters in the task -- the task ``features" -- as well as how to combine these features into a single objective. If they try to do both at once from input designed to teach the full reward function, it is easy to end up with a representation that contains spurious correlations in the data, which fails to generalize to new settings. Instead, our ultimate goal is to enable robots to identify and isolate the causal features that people actually care about and use when they represent states and behavior. Our idea is that we can tune into this representation by asking users what behaviors they consider similar: behaviors will be similar if the features that matter are similar, even if low-level behavior is different; conversely, behaviors will be different if even one of the features that matter differs. This, in turn, is what enables the robot to disambiguate between what needs to go into the representation versus what is spurious, as well as what aspects of behavior can be compressed together versus not. The notion of learning representations based on similarity has a nice parallel in contrastive learning, a self-supervised representation learning technique that maps visually similar data points to similar embeddings, where similarity is defined by a designer through data augmentation heuristics. By contrast, in order to learn the representations that people use, so we can learn their preferences and objectives, we use their definition of similarity. In simulation as well as in a user study, we show that learning through such similarity queries leads to representations that, while far from perfect, are indeed more generalizable than self-supervised and task-input alternatives.