LLM 理论进展

Pre-logit steering adapts a frozen language model to a test-time reward by adding vectors to its final hidden states. Unregularized reward optimization can substantially alter the output distribution and degrade generation quality. We propose Minimally Invasive Steering Vector Optimization (MISVO), which penalizes interventions using the local KL geometry of the induced token distribution. The resulting Fisher quadratic measures distributional sensitivity and admits an analytic gradient computed through matrix--vector products with the frozen language-model head. We derive an exact decomposition of the sequence-level KL gradient into an analytic Fisher term and a suffix score-function term. For a fixed generation horizon, we show that the suffix term is second order in the steering magnitude and that three Fisher surrogates agree with the full KL gradient to first order. MISVO uses the frozen-reference surrogate to optimize position-specific interventions without updating model parameters. Across preference and code-generation tasks on models with approximately 1B--14B parameters, MISVO achieves the highest mean reward in six of seven model--task settings, with diversity and coherence scores close to those of Best-of-N.

Long horizon language model agents continually accumulate reasoning history, increasing context length and inference cost even after earlier decisions have been executed and observed. Unlike static Chain of Thought compression, removing historical reasoning can change future actions and the resulting interaction trajectory. We study when such reasoning can be safely forgotten. We propose Interaction Aware Compression for Long Horizon Reasoning (ICLR), a training free online method that ranks reasoning blocks using frozen proxy entropy while preserving actions, tool calls, and observations. On 260 WorkBuddyBench tasks, ICLR improves average reward from 0.699 to 0.718, while reducing input, output, and cache read tokens by 25.5%, 14.4%, and 33.3%, respectively. Ablations reveal trajectory amplification, where local reasoning deletion produces nonlinear changes in total computation by altering subsequent interaction. Representation probing, activation patching, and controlled trajectory analyses further suggest that historical reasoning becomes more replaceable once task relevant derived state has been reliably externalized into code, files, tool outputs, or environmental feedback. These results characterize agent reasoning as dynamic working state rather than permanent interaction history.

Many-shot in-context learning (ICL) enables large language models (LLMs) to adapt to complex tasks by conditioning on thousands of demonstration examples, but this paradigm shifts the inference efficiency bottleneck to the key-value (KV) cache memory. Due to the linear scaling behavior of the KV cache, storing these intermediate tensors has become a paramount challenge for both online serving and on-device deployment. To address this issue, we propose a novel compression framework, termed MILO, that exploits the low-rank redundancy inherent in many-shot contexts. Specifically, MILO features a block-wise low-rank compression strategy that compresses the KV cache at the block granularity, where each block contains multiple many-shot examples. Furthermore, to handle the heterogeneous context density across different blocks, MILO dynamically allocates rank budgets based on the information entropy, preserving the fidelity of critical blocks while aggressively compressing redundant ones. Experimental results on Qwen2.5 models demonstrate that our method achieves up to 50% reduction in KV cache memory and 1.8x throughput improvement, with negligible performance degradation on classification and reasoning benchmarks, significantly outperforming prior baselines.

Large language models (LLMs) are increasingly deployed in settings where responses must reflect multiple, potentially interacting social norms and human values. Activation steering offers a lightweight alternative to training-based alignment by modifying internal activations at inference time. However, prior human-value steering methods have largely considered values in isolation, while direct composition of multiple directions relies on fixed intervention strengths that cannot respond to the model's evolving internal state. Motivated by this key observation, we introduce AIMES, a framework for adaptive multi-value activation steering. AIMES constructs layer-specific bipolar directions for moral-foundation values and uses intermediate-layer vocabulary readouts as online observers. An observer-guided controller then adapts the strength of each requested value intervention at every decoding step based on its current observed state, without training a separate value-state estimator. Across multiple instruction-tuned model families, value combinations, and intervention depths, we find that multi-value controllability varies across both value combinations and intervention locations. Compared with fixed joint steering and prompt-based steering, AIMES shows depth-dependent advantages that are broadly supported across two independent evaluators, with some variation in the precise depth at which specific control effects emerge. These advantages come with smaller realized activation-space interventions than fixed-joint steering and comparable response quality. Overall, our results suggest that online observer feedback can provide lightweight, state-aware adaptation for single-pass multi-value steering.

Large Language Models (LLMs) consume and produce a single sequence of text; hence, if text can be added to the beginning of the LLM's response, i.e., an output prefix, then all subsequent tokens will be conditioned on it. This output-prefix attack technique is a cheap black-box prompt injection. Prior work has shown this type of attack can reliably jailbreak non-reasoning models. Most reasoning models add an intermediate scratchpad reasoning step before the assistant's final response. The ability to edit this reasoning channel is exposed by some APIs and attack vectors can be leveraged for reasoning injection attacks. We present the first systematic, controlled study that isolates the scratchpad reasoning channel as an output-prefix attack vector, and the first to compare reasoning-only, output-prefix-only and reasoning-plus-output-prefix attacks across both exposed- and hidden-reasoning models. Using a factorial design of 3 prefix types $\times$ 2 reasoning injections over $1{,}800$ test cases drawn from AdvBench, we attack three 2026-era frontier models Gemini 3 Flash Preview, DeepSeek V4 Flash, and Claude Haiku 4.5. We find that injecting malicious reasoning alone is essentially inert ($\approx0%$ attack success), but injecting the same reasoning together with a trivial output prefix raises the attack success rate to as high as $99%$ for some models. For this type of attack we find that contextual prefixes work better than static prefixes; and that susceptibility is dependent on the model.

Layer-wise visual-text similarity in Multimodal Large Language Models (MLLMs) is widely interpreted as evidence that the language model progressively integrates visual content into a shared representation space. This reading rests on the assumption that scalar alignment scores reflect content-level cross-modal interaction. To test this assumption, we apply controlled interventions to the visual stream. Across 13 MLLMs from five families spanning 0.5B to 72B parameters, replacing projector-output visual tokens with Gaussian noise sharply reduces task accuracy, yet four standard scalar measures (CKA, SVCCA, MIR, and the leading principal-angle cosine) fail to consistently separate the corrupted stream from the original. We call this failure the alignment illusion and trace it to the shared language-model pathway: anisotropic MLP down-projections pull visual and text tokens toward common output directions, producing weight-induced alignment. Because this component is essentially one-dimensional, we introduce the principal-angle gap (PA gap), defined as the difference between the top two principal-angle cosines, which separates weight-induced similarity from multi-directional visual structure. Under graded visual corruption, the PA gap tracks task accuracy more consistently than the scalar scores we consider; under a structured but irrelevant image, it further exposes regimes in which internal geometry and task accuracy come apart. Internal visual-text alignment in MLLMs is therefore best read as a geometric diagnostic of the visual stream inside the language model rather than a direct proxy for content-level cross-modal interaction, and is most informative when calibrated by controlled task evidence.

Long-horizon reasoning remains a central challenge for large language models (LLMs) under sparse-reward regimes. We argue that this brittleness arises from two biases induced by complex reasoning spaces: an exploration bias, where models are drawn toward locally plausible but structurally unstable branches, and a compounding bias, where small local deviations accumulate across depth and suppress rare rewards. We introduce Symbolic Closure Analysis (SCA) as a theoretical lens characterizing how branching structures and sparse rewards induce these biases in long-horizon reasoning with local admissibility, and as a design principle for structural priors in less formal reasoning tasks. Motivated by this analysis, we propose SAGE (Structural Admissibility-Guided Exploration), a unified framework that injects structural guidance to alleviate exploration bias and compounding bias in long-horizon reasoning. SAGE combines two complementary structural guidance: algebraic sparsification, which projects locally admissible candidates onto operator-indexed algebraic subspaces to suppress spurious branching and mitigate exploration bias, and hyperbolic structural guidance, which embeds reasoning states into a negatively curved space to provide dense depth-wise signals and mitigate compounding bias. Across 12 benchmarks and 7 model families, SAGE outperforms competitive baselines. In particular, SAGE achieves up to an 8-fold improvement on the Andrews-Curtis problem, an open real-world long-horizon task. Code is available at: https://github.com/Susan571/SAGE-NeurIPS2026.

Property Directed Reachability (PDR) is a prominent algorithm for hardware formal verification. However, bit-level PDR often struggles with datapath-heavy designs because bit-blasting obscures high-level semantics. While word-level PDR addresses this by reasoning over bit-vector and array theories, its performance remains bottlenecked by discovering proof-relevant word-level relations. We propose LLM4PDR, a framework leveraging Large Language Models (LLMs) to guide word-level PDR search through three mechanisms: (1) Predicate Generation, extracting state relationships as candidate predicates during inductive generalization; (2) Clause Generation, producing candidate frame lemmas to accelerate convergence after formal validation; and (3) Assertion Generation, synthesizing helper assertions that strengthen the target property under counterexample-guided refinement. We implement LLM4PDR in the Pono model checker and evaluate it on arithmetic micro-benchmarks, HLS-generated pipelines, open-source RTL components, and hardware model checking competition (HWMCC) instances. Results show LLM-generated guidance improves both solved instances and runtime on datapath-heavy and control-plus-datapath designs. The strongest configuration solves 28 of 33 arithmetic benchmarks, compared to 13 for vanilla Pono and 11 for AVR. On HLS pipelines and open-source RTL, different modes provide complementary speedups: clause guidance is effective for deep pipelines and predicate guidance for bus and memory-controller designs. On HWMCC benchmarks, benefits are instance-dependent, demonstrating notable speedups and timeout avoidance on hard cases. These results suggest that LLM-generated, verifier-checked semantic hints can serve as a practical complement to conventional word-level PDR.

Transformers achieve remarkable performance by jointly learning broad families of tasks during pretraining and adapting to unseen tasks from only a short prompt. Yet a rigorous mathematical and statistical understanding of this phenomenon remains limited. This paper aims to study how Transformers exploit shared cross-task structure and how this structure affects the sample complexity of in-context learning (ICL). Specifically, we characterize task-space complexity through covering numbers under a prescribed metric, thereby quantifying the low-dimensional cross-task structure without requiring an explicit parametric representation. The resulting cover provides a set of anchor functions, which we use to introduce a task-identification-and-evaluation procedure: context observations localize an unseen task among the anchor functions, and the response at a query is predicted by aggregating the corresponding anchor function query evaluations. For approximation, we explicitly construct a Transformer with Softmax attention to approximate this procedure. For generalization, we derive an error bound that separates the effects of the number of pretraining tasks and the prompt length. The scaling with respect to the number of pretraining tasks is governed by the intrinsic dimensions of the task space and input domain; once sufficiently many tasks are available, the dependence on the prompt context length becomes dimension-free. To the best of our knowledge, this is the first work to quantify cross-task complexity for general nonlinear task families and explicitly construct a Transformer that exploits their low-dimensional structure to perform ICL. Our theory provides a quantitative explanation of how joint pretraining across related tasks improves in-context generalization.

Step Law gives power-law formulas for the optimal peak learning rate eta* and batch size B* when pre-training language models. It was calibrated on models between 59M and 1B parameters; the small-model regime N < 59M was never tested empirically by its authors. This regime matters for single-GPU training, interpretability research, educational experiments, and settings where larger models are infeasible on memory or cost grounds. We test whether Step Law transfers to small language models. We consider three outcomes: H1, the original coefficients work directly; H2, the power-law form holds but with different coefficients; and H3, a power law does not describe the optima in this regime. All experiments use a single nanoGPT/TinyStories pipeline with a 2048-token BPE vocabulary, AdamW, and a warmup-cosine schedule. The optimum for each (N, D) cell is extracted from the loss surface L(eta, B) via a local quadratic approximation in log-log coordinates over the smoothed training loss. The final dataset contains 29 unique (N, D) cells and 935 analysis-ready runs. The main refit uses 25 cells (815 runs) in the working range 4 <= D/N <= 600. On the pooled data we accept H2: the functional form is preserved, but the coefficients differ from the original. We obtain eta*(N, D) = 0.0985 N^(-0.508) D^(0.238) (R^2 = 0.834) and B*(D) = 3.6 x 10^(-4) D^(0.931) (R^2 = 0.950). Step Law's structural claim that B* is independent of N is reproduced (p = 0.87), but the growth of B* with D is nearly twice as steep as in the original work. Direct transfer of Step Law systematically overestimates the optimal learning rate: the median ratio eta_SL / eta* is approximately 4.0x, with a range of 2.4x to 6.6x.

In-Context Learning (ICL) has become a cornerstone of modern LLM deployment. However, existing ICL post-training methods have a critical blind spot: they excel at extracting patterns from demonstrations while often neglecting context authority, the ability to determine whether contextual information should govern the final answer. To benchmark this capability, we introduce FakeContextBench, which contains pseudoscientific claims across seven domains. Our evaluation of commercial and open-source models shows that large-scale pre-training alone is insufficient for reliable context-authority discrimination. Moreover, prevalent ICL fine-tuning methods can increase susceptibility to misleading context, reducing reality accuracy by up to 14.95 percentage points relative to the base model. To address this trade-off, we propose Jurisdiction In-Context Learning (J-ICL), a post-training framework that incorporates context validation into the training objective. Across four model backbones, J-ICL improves ICLEval by an average of 5.84 percentage points and reality accuracy by 9.20 points over the corresponding base models. It also raises the Reality Rate by an average of 18.09 points relative to MetaICL and Symbol Tuning. These results demonstrate that ICL capability and resistance to deceptive context can be improved together. The benchmark is available at https://github.com/peilin717/FakeContext-Bench.

In-context learning (ICL) enables large vision-language models (LVLMs) to perform tasks by following patterns from in-context examples, yet its potential to amplify societal biases remains underexplored. We systematically investigate how ICL influences gender bias in LVLMs through VL-BICLE, an evaluation framework comprising six ICL settings, three tasks, and four datasets. Our experiments on six LVLMs reveal that gendered ICL demonstrations act as a directional force, shifting model bias toward the demonstrated gender through a cross-gender mechanism that disproportionately degrades performance on the opposite gender. This effect appears in image captioning and pronoun prediction but not in visual question answering, indicating that gendered ICL influences bias only when the task output involves gendered language. Similarity-based retrieval methods inherit the training pool's gender imbalance and offer no debiasing advantage, while standard quality metrics remain blind to these bias shifts. To mitigate this bias, we replace real in-context images with synthetic ones from stable diffusion models while keeping captions unchanged. This simple intervention reduces gender bias without degrading caption quality.

In this paper, we introduce a gradient-based head attribution strategy where the Token-level Max-Margin loss is backpropagated to the attention maps. This framework enables a large-scale causal analysis of attention heads, making it suitable for LLMs. We evaluate our method on the task of disambiguation in Context-aware Machine Translation, where we analyze 50 phenomena across 4 models and 4 language directions. We empirically show the alignment of our method with the effects of increasing the attention scores of token-to-token relations on three models and two language directions, ensuring the robustness of our method. Our analysis reveals the presence of the "general-purpose" attention heads that improve the model's performance when attending to different relations. We find that the average attention a head assigns to a relation does not necessarily relate to the model's performance, which suggests that the models developed redundancies during training in terms of the head functions.

Despite the importance of reinforcement learning with verifiable rewards (RLVR), the extent to which it can learn new reasoning capabilities remains debated. Here we study the optimization landscape of RLVR on algorithmic tasks, such as iterated group and quasigroup multiplication. To this end, we map entropy-regularized RLVR over myopic tabular policies onto an energy-based (spin-glass) model over deterministic policies. This mapping upper-bounds what RLVR can achieve, and lets us rigorously characterize the landscape in this tabular setting. We show, both theoretically and experimentally, that for a wide class of models and tasks with uncorrelated inputs, this landscape is benign, containing no local minima that could trap RLVR training. Rather, the practical difficulty of these tasks appears to stem, at least in part, from issues such as diffusive barriers and gradient-estimation error in traversing the landscape. These are genuine obstacles that can prevent a solution from being found, but they are distinct from the landscape itself being rugged. We show that these obstacles can often be mitigated through the choice of entropy regulator. Consistent with this theory, we find that a transformer trained from scratch, using only last-token rewards, successfully learns an algorithmic chain of thought for iterated non-Abelian group multiplications.

Vision-language models (VLMs) achieve strong visual reasoning performance, yet subtle changes from routine image capture and processing can alter their reasoning trajectories even when images appear nearly identical. In long-horizon generation, the resulting activation shifts may accumulate across decoding steps, progressively altering reasoning tokens and ultimately changing the final answer, a phenomenon referred to as answer flips. To address this instability, we propose FlipDir (Flip-Direction Steering), a training-free inference-time method that estimates a low-rank flip-inducing activation subspace from contrastive pairs of original and answer-flipping inputs and selectively steers hidden states during decoding. A margin-based gate limits subspace attenuation to uncertain decoding steps, recovering original predictions while preserving stable ones. To evaluate robustness beyond accuracy or consistency on fixed test sets, we introduce VisFlip, a benchmark framework that constructs evaluation groups for a target model and visual variation setting to separately assess recovery of original predictions and preservation of stable ones. VisFlip spans nine dataset-variation combinations across scientific reasoning, robot-scene understanding, and medical VQA, covering subtle visual variations common in each domain. Experiments across 18 settings demonstrate that FlipDir consistently outperforms existing methods on the combined recovery and preservation metric. We will make our code publicly available.

System prompts are the primary lever practitioners use to control language model behavior, yet what they actually do to the computation inside the transformer remains poorly understood. Across 17 instruction-tuned models spanning 8 architecture families and 1.5B to 72B parameters, we use Centered Kernel Alignment (CKA) to compare layer-wise representations under 20 system prompts in five functional categories. Effects are layer-selective and instruction-type-dependent: persona and formatting instructions deeply restructure intermediate representations, while safety instructions barely move them, producing changes statistically indistinguishable from a minimal baseline. Restrictive safety instructions and explicitly permissive ones ("you have no restrictions") engage near-identical computational pathways (mean CKA correlation 0.997), and this persists at commercial scale, where safety penetration remains below 10% even at 70B-72B. A linear probing baseline exposes the mechanism: the model encodes prompt category at every layer but restructures its computation only at a small subset, so the prompt is reliably "seen" but, for safety, not deeply "acted upon." Causal activation patching confirms these layers mediate behavioral change, and representational depth predicts behavioral effect size across the full 17-model cohort (Spearman rho = 0.761, p < 0.001). The findings provide a mechanistic explanation for the persistent jailbreak vulnerability of system-prompt-based safety. Code: https://github.com/Usama1002/system-prompt-illusion-cka

Reinforcement learning (RL) has emerged as a key paradigm for improving the reasoning capabilities of large language models (LLMs). However, existing reward systems, such as rule-based and reward-model-based, often exhibit issues such as unstable optimization and reward hacking. In this work, we revisit the general reasoning of LLMs from a geometric perspective, conceptualizing it as a coupled manifold composed of three interdependent sub-manifolds: logical deduction, evaluation, and representation. Based on this perspective, response generation in RL can be interpreted as a decoupling process from the evaluation manifold, while reward estimation corresponds to a decoupling process from the logical deduction manifold. The limitations of rule-based and reward-model RL systems can be geometrically interpreted as the mismatch of policy-reward manifolds during RL process. To address the aforementioned misalignment, we propose Decoupling and Coupling Reinforcement Learning (DCRL) framework, which incorporates two key components: (1) a syllogistic logic-based prompt evolution mechanism that dynamically refines reward rubrics to enhance the expressiveness of the reward manifold; and (2) a policy-reward re-coupling mechanism that jointly updates the reward and policy models, ensuring consistent evaluation and mitigating manifold mismatch during training. Theoretical analysis and extensive experiments across multiple reasoning domains demonstrate that DCRL consistently outperforms both rule-based and reward-model baselines. Notably, a Qwen3-4B model trained under DCRL surpasses a Qwen3-32B baseline and approaches the performance of a Qwen3-235B model, highlighting superior effectiveness and generalization in RL.

Vision-language models (VLMs) offer a promising approach to long-tail autonomous driving, but existing driving datasets provide limited supervision for connecting decision-critical visual evidence with reasoning and planning. We introduce AnchorReasoning, a visually grounded reasoning dataset built on WOD-E2E, containing 416,119 annotated frames and 395,379 decision-critical elements across four major categories and 19 fine-grained types. Each frame is organized as a visually grounded chain-of-thought (VG-CoT) that links decision-critical element identification and localization, element attributes and implications, driving-action rationale, and action and trajectory planning. We further develop a curriculum supervised fine-tuning strategy that progressively learns these hierarchical capabilities, together with an object-size-aware grounding metric for evaluating localization quality. Experiments across eight general-purpose, embodied-AI, and AV-specific backbones show that VG-CoT supervision improves grounded reasoning and trajectory prediction. Across models, 5-s ADE and FDE decrease by 7.84 and 11.86, while RFS Frame and Cluster improve by 1.66 and 1.70. These gains are achieved with 18.5 fewer reasoning tokens and 0.32 s/frame lower inference latency on average, demonstrating the value of visually grounded, decision-focused supervision for VLM reasoning and planning in long-tail autonomous driving.

Large Language Model (LLM) agents struggle to scale safely when exposed to vast enterprise toolsets. Providing an agent with access to every internal tool leads to oversized context windows, degraded tool selection, and severe governance vulnerabilities - as system policies defined purely in prompts remain probabilistic advice rather than hard constraints. Existing mitigations, such as multi-agent domain delegation, decentralize audit logs and fail to guarantee policy compliance across sessions. We introduce skilder, a framework that packages capabilities into roles: bundles of skills, tools, and instructions, together with the limits that bound them. An agent begins with a minimal role catalog, learns the roles a task requires, and receives each role's skills, instructions, and tools through a single MCP server. Because tools reach the agent only inside learned skills, the same server enforces the scope of what was learned deterministically. We evaluate skilder against flat-context tool selection and multi-agent orchestration across 13 tasks using six models (10 runs each). Our results show that, when models completed discovery and issued a governed call, the skilder simulated authorization layer enforced governance boundaries: no unauthorized tool call or parameter violation (e.g., a spending-limit breach) executed. Aggregate task pass rates also reflect whether each model followed the discovery protocol and satisfied response-quality checks; those misses are not authorization failures. Furthermore, by allowing agents to dynamically acquire cross-role capabilities mid-task, skilder preserves problem-solving flexibility while providing hard system-level enforcement.

Language models typically construct attention values from contextual hidden states, even when some of their content may be reusable across contexts. We investigate whether token-indexed memory can replace the dedicated value projection when complemented by contextual information. We propose Memory Attention (MA), which forms values by combining layer-specific token memory with contextual keys. The memory supplies token-specific representations, while the keys preserve context dependence. At inference, normalization can be folded into the memory tables, reducing value construction to lookup and addition. Token-indexed retrieval also enables CPU offloading with prefetching, reducing GPU parameter storage. Under matched training token budgets and with additional memory parameters, experiments across attention configurations show improved language modeling and average downstream performance.

The key-value (KV) cache of autoregressive transformers can be viewed as a fourth-order tensor spanning attention heads, tokens, features, and grouped layers. We measure the singular-value spectra of all four mode unfoldings on Mistral-7B-v0.3 and LLaMA-2-13B and compare four standard tensor decompositions: Tucker, CP, tensor train, and t-SVD, at matched storage. The spectra partition the four axes into two classes. The token and feature modes carry low-rank structure, particularly for keys. The head and layer modes are nearly full-rank and resist compression at any practical error level. Among the four decompositions, Tucker achieves the lowest reconstruction error at every compression ratio from $2\times$ to $5\times$, because it can leave the full-rank modes untouched. Comparisons with two-dimensional unfolding baselines show that the preferred representation differs between keys and values: 2D methods achieve lower key error, while four-way Tucker achieves lower value error at matched storage. A mode-pinning theorem certifies the full-rank preservation from the measured spectra alone. Two further spectral properties affect the compressible modes without touching the full-rank ones: values reach a higher error floor than keys at every ratio, and post-RoPE keys lose $41%$ - $64%$ of their pre-RoPE compressibility on both models.

Newton-Schulz (NS) iteration has recently been used in the Muon optimizer to transform update matrices during the training of large language models. Motivated by its spectral effect, we investigate applying NS directly to Transformer attention representations. We introduce Newton-Schulz Attention (NS-Attn.), a parameter-free transformation applied to the output of each attention head. Each head output is arranged as a feature-by-token matrix and normalized by its Frobenius norm. We then apply a finite NS polynomial step and restore the original norm. The objective is to reduce spectral concentration and increase effective rank before standard head merging and output projection. Across ViT and Swin on CIFAR-10 and CIFAR-100, NS-Attn. improves final-epoch accuracy in all 12 matched-seed comparisons, with mean gains of 0.25--0.83 percentage points. ViT ablations show higher mean accuracy with one iteration than with two. Spectral analysis further shows reduced leading-eigenvalue concentration and increased effective rank. These gains incur additional inference latency.

In this paper, we are concerned with matrices formed by block-diagonal factors interleaved with fixed permutations — a flexible family of structured matrices. This class has recently drawn interest in deep learning architectures for its balanced expressivity-efficiency trade-off, yet efficient computational strategies for working with it remain to be found. We approach this problem through Riemannian geometry and examine under what conditions this class admits a smooth manifold structure. For the practically important case of orthogonal two-factor matrices, we derive the essential Riemannian tools and propose efficient algorithms for their implementation. The algorithms leverage automatic differentiation, support parameter sharing within each factor, and avoid explicit dense matrix construction. We test them within the Riemannian optimization framework on the best matrix approximation problem and for parameter-efficient fine-tuning of large language models. Beyond the two-factor setting, we study the geometric and matrix-theoretic properties of factorizations with a larger number of block-diagonal factors.

Recurrent-depth language models, such as looped Transformers, repeatedly apply shared network blocks to refine latent representations without generating explicit intermediate reasoning tokens. However, each step recomputes full attention over the entire context, repeating costly global routing. We study how attention routing evolves across recurrent depth and find a consistent separation in convergence timescales: attention support and distributions stabilize substantially earlier than hidden states and attention outputs. This suggests two stages of recurrent inference: early discovery of a sparse working set, followed by representation refinement over largely stable routing support. Motivated by this finding, we introduce WISE (Working-set Inference with Support Exploitation), a training-free method that uses unrestricted attention during early recurrent steps to discover a block-structured working set, then reuses its support in later steps while keeping attention weights and recurrent refinement dynamic. Controlled interventions show that multi-step discovery yields more effective working sets than first-step selection, and that support reuse better preserves model behavior than more restrictive forms of attention reuse. Across multi-hop QA benchmarks, WISE largely preserves full-attention performance. Matched context-scaling experiments reveal an increasingly favorable quality-efficiency tradeoff as routing support becomes sparser with longer contexts. A sparse-attention implementation achieves up to a 1.76x late-step attention speedup over native FlashAttention at 4K context. Code: https://github.com/tbn5pj/WISE_code.

Diffusion language models generate text through iterative denoising, exposing intermediate trajectories before final answers are produced. We identify a recurring reasoning failure, stable-but-wrong lock-in, where an answer stabilizes early around an incorrect value while substantial denoising remains. Surface-level decoding signals such as confidence, entropy, margin, and answer stability are insufficient to reliably distinguish correct from erroneous lock-in. We formulate selective reasoning repair as a lightweight test-time planning problem and propose LOCKR, a hidden-state trajectory-guided planner that decides when to allocate additional computation, expands a structured set of targeted repair branches, and selects the most promising continuation using trajectory-aware verification. Across two diffusion language models and three mathematical reasoning benchmarks, hidden-state trajectories consistently outperform surface signals and single hidden snapshots for both wrong-lock-in detection and repair selection. On natural evaluation distributions, LOCKR yields absolute accuracy gains of 2.21--5.37 percentage points across all five evaluated settings, with repair rates ranging from 22% to 41%. These results establish hidden diffusion trajectories as actionable signals for selective test-time reasoning repair.

While various multimodal methods combining imaging and tabular data for Alzheimer's disease (AD) diagnosis were proposed, they are often limited in generalization across cohorts. In-context learning (ICL) has demonstrated excellent generalization performances and high flexibility in foundational tabular models such as TabPFN. To extend TabPFN's ICL to multimodal AD analysis, the main obstacle is that TabPFN is meta-trained on synthetic tabular priors that do not naturally match the statistical structure of image-derived features. We propose M$^2$PFN, an end-to-end framework that turns this tabular foundation model into a multimodal AD predictor. M$^2$PFN (i) performs differentiable inference through TabPFN's transformer, back-propagating task gradients into 3D-MRI and tabular encoders; (ii) aligns the two modalities into a shared subspace, via disentanglement and a contrastive objective, matched to the ICL engine's prior; and (iii) folds in a frozen tabular-only prediction through a learnable gated shortcut. Because the ICL engine stays frozen, its in-context mechanism is preserved for test-time generalization, while end-to-end training shapes the encoders into features it can exploit. On ADNI ($n=2240$, three-class CN/MCI/AD), M$^2$PFN attains $65.55%$ macro-F1 and $82.21%$ macro-AUC, surpassing a comprehensive set of unimodal and multimodal baselines. By swapping only the head for a TabPFN regressor, the same architecture regresses baseline MMSE on a $1250$-subject sub-cohort to test MAE $1.743$, outperforming every multimodal baseline. On two external cohorts (OASIS-3 and SCAN) with no retraining, M$^2$PFN achieves the best AUC and the lowest MMSE MAE across all baselines, and transfers even when the cognitive instrument changes.

Reordering a set of mathematical rules without changing its meaning should preserve the correct answer, but must a model's internal representations stay invariant too? We investigate this question using synthetic multi-step function-composition problems, each presented under multiple rule orderings with the same correct answer. We measure accuracy and permutation signal-to-noise ratio (SNR), which quantifies how distinctly ordering patterns are represented relative to variation across problem instances. Across 16 language models ranging from 1B to 8B parameters, we find a pattern: models that solve reordered problems more accurately represent different rule orderings more distinctly. Layer-averaged permutation SNR is positively rank-correlated with accuracy in every synthetic setting we evaluate, with Spearman correlations reaching 0.86. These findings highlight a distinction between answer invariance and representation invariance: successful mathematical rule composition can accompany distinct internal representations between equivalent rule orderings. This motivates distinguishing answer invariance from representation invariance, and offers a representational perspective on mathematical reasoning beyond answer accuracy alone.

Diffusion Language Models (DLMs) have attracted significant attention for their strong reasoning ability. However, under a bidirectional attention mechanism, DLMs operate over an exponentially large exploration space compared to autoregressive models (ARMs), making it challenging to focus on reasoning-guiding tokens under random masking. We define causal shortcuts as token chains that cover the full sequence and provide explicit guidance towards correct reasoning trajectories. We analyze the effects of causal shortcuts on the reasoning accuracy and convergence speed of DLMs, and find that they largely improve answer convergence efficiency and generation accuracy. Motivated by this, we propose a Causal Shortcut Learning (CSL) Framework for DLMs. Specifically, we introduce a step-by-step token extraction procedure to extract causal shortcuts from data, and apply parallel prioritized masking on these tokens during training to enable efficient and accurate convergence to correct answers via causal shortcuts. Extensive experiments across multiple reasoning benchmarks and two base models demonstrate that CSL consistently outperforms existing SFT-variant baselines, achieving an average improvement of $1.92%$ over SFT-only models, and up to $4.20%$ on MATH-500. The code is available at the \href{https://github.com/ZJUDianJin/Causal-Shortcuts-Learning}{https://github.com/ZJUDianJin/Causal-Shortcuts-Learning

Large language models can interpret natural lan- guage, yet robust decisions remain challenging. Jev-like models expose structured choices, but these interfaces do not directly provide numeri- cal values at a requested precision. We propose NUMERICJEV, a training-free numerical decod- ing algorithm that enables numerical output from any LLM with a Jev-like structured-choice in- terface. Surprisingly, on our arithmetic bench- mark, it outperforms direct selection from a can- didate list containing the correct answer by 2.93 percentage points (Figure 1). Our motivation comes from the observation that numerical range selection is itself a decision problem that Jev- like LLMs can address. NUMERICJEV recur- sively refines a range through a multiway deci- sion tree while retaining the original question in context, without parameter updates or hidden- state access. On a 100-value grid, a ten-way tree requires only two decision rounds. Range- normalized MAE is 1.84% versus 5.18% for di- rect choice. A separate three-date historical- index study yields 4.58% mean relative recall er- ror and 0% readout error when the value is sup- plied. Code is available at https://github. com/Bring-AI/jev-numeric.

With the rising computational and monetary costs of training large language models (LLMs), checkpointing---periodically storing model states for recovery---becomes essential for fault tolerance. Conventional checkpointing entails a severe trade-off between checkpoint frequency (I/O overhead) and computational recovery (recovery time). State-of-the-art approaches mitigate this cost through pipelining checkpoint I/Os, differential checkpointing, or in-memory persistence, yet none leverage the distinct characteristics of LLM training dynamics, where model weight updates are non-uniformly distributed across transformer layers. This observation implies that saving all weights each time might not be efficient. Inspired by this observation, we present LayerCheck, a layer-wise adaptive checkpointing framework that selectively persists layers whose updates exceed a threshold. This design avoids periodic I/O bursts by distributing layer-wise checkpoint writes over time, resulting in smoother and more balanced I/O profiles. Upon recovery, LayerCheck reconstructs a mixed-timestamp composite model state by aggregating the most recently persisted versions of each layer together with their matching optimizer states. Under a bounded per-layer staleness guard, this introduces a controlled perturbation: under standard Adam assumptions it adds a bounded staleness term, and empirically the post-restart loss deviates from the failure-free trajectory by at most 0.54%. Empirical results on multiple open-source LLMs with different datasets further demonstrate that recovered models preserve the original convergence behavior and accuracy while substantially reducing checkpoint overheads. Specifically, LayerCheck achieves up to 22.6x reduction in total checkpoint size and 1.31x reduction in end-to-end training time compared to state-of-the-art systems, significantly lowering the cost of checkpointing.