Daily Papers

Guardrail Classifiers defend production language models against harmful behavior, but although results seem promising in testing, they provide no formal guarantees. Providing formal guarantees for such models is hard because "harmful behavior" has no natural specification in a discrete input space: and the standard epsilon-ball properties used in other domains do not carry semantic meaning. We close this gap by shifting verification from the discrete input space to the classifier's pre-activation space, where we define a harmful region as a convex shape enclosing the representations of known harmful prompts. Because the sigmoid classification head is monotonic, certifying the worst-case point is sufficient to certify the entire region, yielding a closed-form soundness proof without approximation in O(d) time. To formally evaluate these classifiers, we propose two constructions of such regions: SVD-aligned hyper-rectangles, which yield exact SAT/UNSAT certificates, and Gaussian Mixture Models, which yield probabilistic certificates over semantically coherent clusters. Applying this framework to three author-trained Guardrail Classifiers on the toxicity domain, every hyper-rectangle configuration returns SAT, exposing verifiable safety holes across all classifiers, despite seemingly high empirical metrics. Probabilistic GMM certificates also expose a divergent structural stability in how these models represent harm. While GPT-2 and Llama-3.1-8B maintain robust coverage of 90% and 80% across varying boundaries, BERT's safety guarantees prove uniquely volatile. This 'coverage collapse' to 55% at the optimal threshold reveals a sparsely populated safety margin in BERT, which only achieves full coverage by adopting an extremely conservative pessimistic threshold. These approaches combined, provide new insights on how effective Guardrail Classifiers really are, beyond traditional red-teaming.

Gradient-based preference optimization methods for large language model (LLM) alignment suffer from preference collapse, converging to narrow behavioral modes while neglecting preference diversity. We introduce EvoPref, a multi-objective evolutionary algorithm that maintains populations of Low-Rank Adaptation (LoRA) adapters optimized across helpfulness, harmlessness, and honesty objectives using Non-dominated Sorting Genetic Algorithm II (NSGA-II) selection with archive-based diversity preservation. Our primary contribution is demonstrating that population-based methods discover substantially more diverse alignments than gradient descent. On standard benchmarks, EvoPref improves preference coverage by 18% (median 82.5% vs. 70.0% for ORPO, p<0.001, Wilcoxon, n=30) and reduces collapse rates by 47% (11.0% vs. 20.6%, p<0.001), while achieving competitive alignment quality (median 75.5% RewardBench vs. 75.0% for ORPO, p<0.05). We provide theoretical motivation extending recent multi-objective evolutionary algorithm (MOEA) runtime analysis (Dang et al., 2025) suggesting why archive-based methods escape collapse more effectively than single-trajectory optimization. Comprehensive comparisons against MOEA/D, SMS-EMOA, CMA-ES, and gradient baselines (DPO, IPO, KTO, ORPO) with rigorous statistical testing (Friedman with Holm correction, Vargha-Delaney effect sizes, median with IQR) confirm that multi-objective selection with diversity preservation is essential. This work establishes evolutionary optimization as a principled paradigm for diverse LLM alignment.

Reinforcement Learning with Verifiable Rewards (RLVR) is increasingly viewed as a tree pruning mechanism. However, we identify a systemic pathology termed Recursive Space Contraction (RSC), an irreversible collapse driven by the combined dynamics of positive sharpening and negative squeezing, where the sampling probability of valid alternatives vanishes. While Kullback-Leibler (KL) regularization aims to mitigate this, it imposes a rigid Shape Matching constraint that forces the policy to mimic the reference model's full density, creating a gradient conflict with the sharpening required for correctness. We propose Anchored Policy Optimization (APO), shifting the paradigm from global Shape Matching to Support Coverage. By defining a Safe Manifold based on the reference model's high-confidence support, APO permits aggressive sharpening for efficiency while selectively invoking a restorative force during error correction to prevent collapse. We theoretically derive that APO serves as a gradient-aligned mechanism to maximize support coverage, enabling an Elastic Recovery that re-inflates valid branches. Empirical evaluations on mathematical benchmarks demonstrate that APO breaks the accuracy-diversity trade-off, significantly improving Pass@1 while restoring the Pass@K diversity typically lost by standard policy gradient methods.

Vision-Language Models (VLMs) have demonstrated remarkable capabilities but suffer from significant computational overhead during inference. While visual token pruning offers a promising solution, existing methods predominantly rely on initial attention scores. This single-metric paradigm presents a critical flaw: high attention scores inherently collapse onto semantically similar regions, thereby severely reducing feature diversity and discarding vital contextual details. To address this, we introduce Structure-to-Semantics (STS), a novel two-stage visual token pruning framework that explicitly decouples the pruning process. The first stage employs a repulsion-based sampling mechanism to maximize spatial and structural diversity. The second stage leverages instruction-aware cross-attention to precisely filter out prompt-irrelevant tokens. This two-stage synergy constitutes the core of STS, first ensuring geometric coverage and then refining the retained tokens according to semantic relevance. Extensive evaluations demonstrate that STS mitigates the redundancy caused by attention-based selection, improving both structural diversity and fine-grained task alignment of the preserved visual tokens.

We study whether an aperiodic hierarchy can provide a structural advantage for lossless compression over periodic alternatives. We show that Fibonacci quasicrystal tilings avoid the finite-depth collapse that affects periodic hierarchies: usable n-gram lookup positions remain non-zero at every level, while periodic tilings collapse after O(log p) levels for period p. This yields an aperiodic hierarchy advantage: dictionary reuse remains available across all scales instead of vanishing beyond a finite depth. Our analysis gives four main consequences. First, the Golden Compensation property shows that the exponential decay in the number of positions is exactly balanced by the exponential growth in phrase length, so potential coverage remains scale-invariant with asymptotic value Wvarphi/5. Second, using the Sturmian complexity law p(n)=n+1, we show that Fibonacci/Sturmian hierarchies maximize codebook coverage efficiency among binary aperiodic tilings. Third, under long-range dependence, the resulting hierarchy achieves lower coding entropy than comparable periodic hierarchies. Fourth, redundancy decays super-exponentially with depth, whereas periodic systems remain locked at the depth where collapse occurs. We validate these results with Quasicryth, a lossless text compressor built on a ten-level Fibonacci hierarchy with phrase lengths {2,3,5,8,13,21,34,55,89,144}. In controlled A/B experiments with identical codebooks, the aperiodic advantage over a Period-5 baseline grows from 36{,}243 B at 3 MB to 11{,}089{,}469 B at 1 GB, explained by the activation of deeper hierarchy levels. On enwik9, Quasicryth achieves 225{,}918{,}349 B (22.59%), with 20{,}735{,}733 B saved by the Fibonacci tiling relative to no tiling.

Fine-tuning diffusion models via online reinforcement learning (RL) has shown great potential for enhancing text-to-image alignment. However, since precisely specifying a ground-truth objective for visual tasks remains challenging, the models are often optimized using a proxy reward that only partially captures the true goal. This mismatch often leads to reward hacking, where proxy scores increase while real image quality deteriorates and generation diversity collapses. While common solutions add regularization against the reference policy to prevent reward hacking, they compromise sample efficiency and impede the exploration of novel, high-reward regions, as the reference policy is usually sub-optimal. To address the competing demands of sample efficiency, effective exploration, and mitigation of reward hacking, we propose Gated and Adaptive Regularization with Diversity-aware Optimization (GARDO), a versatile framework compatible with various RL algorithms. Our key insight is that regularization need not be applied universally; instead, it is highly effective to selectively penalize a subset of samples that exhibit high uncertainty. To address the exploration challenge, GARDO introduces an adaptive regularization mechanism wherein the reference model is periodically updated to match the capabilities of the online policy, ensuring a relevant regularization target. To address the mode collapse issue in RL, GARDO amplifies the rewards for high-quality samples that also exhibit high diversity, encouraging mode coverage without destabilizing the optimization process. Extensive experiments across diverse proxy rewards and hold-out unseen metrics consistently show that GARDO mitigates reward hacking and enhances generation diversity without sacrificing sample efficiency or exploration, highlighting its effectiveness and robustness.

We present Branch-Train-Merge (BTM), a communication-efficient algorithm for embarrassingly parallel training of large language models (LLMs). We show it is possible to independently train subparts of a new class of LLMs on different subsets of the data, eliminating the massive multi-node synchronization currently required to train LLMs. BTM learns a set of independent expert LMs (ELMs), each specialized to a different textual domain, such as scientific or legal text. These ELMs can be added and removed to update data coverage, ensembled to generalize to new domains, or averaged to collapse back to a single LM for efficient inference. New ELMs are learned by branching from (mixtures of) ELMs in the current set, further training the parameters on data for the new domain, and then merging the resulting model back into the set for future use. Experiments show that BTM improves in- and out-of-domain perplexities as compared to GPT-style Transformer LMs, when controlling for training cost. Through extensive analysis, we show that these results are robust to different ELM initialization schemes, but require expert domain specialization; LM ensembles with random data splits do not perform well. We also present a study of scaling BTM into a new corpus of 64 domains (192B whitespace-separated tokens in total); the resulting LM (22.4B total parameters) performs as well as a Transformer LM trained with 2.5 times more compute. These gains grow with the number of domains, suggesting more aggressive parallelism could be used to efficiently train larger models in future work.

Sampling from pretrained diffusion and flow-matching models typically requires many forward passes to generate diverse and high-fidelity images. Existing distillation methods often rely on multiple auxiliary networks, carefully designed training stages, or complex optimization pipelines. In this work, we revisit the recently proposed Drifting Model objective and show that a single drifting loss can be directly used to simplify one step distillation. A key observation is that the pretrained diffusion teacher itself already provides a strong representation space. Unlike the original Drifting Model, which relies on an additional pretrained feature extractor, we use intermediate hidden states of the pretrained teacher model as the feature representation. This removes the need for training or introducing an extra representation network while preserving a semantically meaningful feature geometry for drifting. Furthermore, we introduce a lightweight mode coverage loss to mitigate mode collapse during distillation and encourage the student generator to cover diverse teacher-supported regions. Extensive experiments on ImageNet and SDXL demonstrate that our method achieves efficient one step generation with competitive image quality and diversity, achieving FID scores of 1.58 on ImageNet-64times64 and 18.4 on SDXL, while substantially simplifying the overall distillation framework.

We introduce Synthetic Alignment data Generation for Safety Evaluation and Red Teaming (SAGE-RT or SAGE) a novel pipeline for generating synthetic alignment and red-teaming data. Existing methods fall short in creating nuanced and diverse datasets, providing necessary control over the data generation and validation processes, or require large amount of manually generated seed data. SAGE addresses these limitations by using a detailed taxonomy to produce safety-alignment and red-teaming data across a wide range of topics. We generated 51,000 diverse and in-depth prompt-response pairs, encompassing over 1,500 topics of harmfulness and covering variations of the most frequent types of jailbreaking prompts faced by large language models (LLMs). We show that the red-teaming data generated through SAGE jailbreaks state-of-the-art LLMs in more than 27 out of 32 sub-categories, and in more than 58 out of 279 leaf-categories (sub-sub categories). The attack success rate for GPT-4o, GPT-3.5-turbo is 100% over the sub-categories of harmfulness. Our approach avoids the pitfalls of synthetic safety-training data generation such as mode collapse and lack of nuance in the generation pipeline by ensuring a detailed coverage of harmful topics using iterative expansion of the topics and conditioning the outputs on the generated raw-text. This method can be used to generate red-teaming and alignment data for LLM Safety completely synthetically to make LLMs safer or for red-teaming the models over a diverse range of topics.

LLM-as-judge evaluation has become the de facto standard for scaling model assessment, but the practice is statistically unsound: uncalibrated scores can invert preferences, naive confidence intervals on uncalibrated scores achieve near-0% coverage, and importance-weighted estimators collapse under limited overlap despite high effective sample size (ESS). We introduce Causal Judge Evaluation (CJE), a framework that fixes all three failures. On n=4,961 Chatbot Arena prompts (after filtering from 5k), CJE achieves 99% pairwise ranking accuracy at full sample size (94% averaged across configurations), matching oracle quality, at 14x lower cost (for ranking 5 policies) by calibrating a 16x cheaper judge on just 5% oracle labels (~250 labels). CJE combines three components: (i) AutoCal-R, reward calibration via mean-preserving isotonic regression; (ii) SIMCal-W, weight stabilization via stacking of S-monotone candidates; and (iii) Oracle-Uncertainty Aware (OUA) inference that propagates calibration uncertainty into confidence intervals. We formalize the Coverage-Limited Efficiency (CLE) diagnostic, which explains why IPS-style estimators fail even when ESS exceeds 90%: the logger rarely visits regions where target policies concentrate. Key findings: SNIPS inverts rankings even with reward calibration (38% pairwise, negative Kendall's tau) due to weight instability; calibrated IPS remains near-random (47%) despite weight stabilization, consistent with CLE; OUA improves coverage from near-0% to ~86% (Direct) and ~96% (stacked-DR), where naive intervals severely under-cover.

Text-to-image (T2I) generative models achieve impressive visual fidelity but inherit and amplify demographic imbalances and cultural biases embedded in training data. We introduce T2I-BiasBench, a unified evaluation framework of thirteen complementary metrics that jointly captures demographic bias, element omission, and cultural collapse in diffusion models - the first framework to address all three dimensions simultaneously. We evaluate three open-source models - Stable Diffusion v1.5, BK-SDM Base, and Koala Lightning - against Gemini 2.5 Flash (RLHF-aligned) as a reference baseline. The benchmark comprises 1,574 generated images across five structured prompt categories. T2I-BiasBench integrates six established metrics with seven additional measures: four newly proposed (Composite Bias Score, Grounded Missing Rate, Implicit Element Missing Rate, Cultural Accuracy Ratio) and three adapted (Hallucination Score, Vendi Score, CLIP Proxy Score). Three key findings emerge: (1) Stable Diffusion v1.5 and BK-SDM exhibit bias amplification (>1.0) in beauty-related prompts; (2) contextual constraints such as surgical PPE substantially attenuate professional-role gender bias (Doctor CBS = 0.06 for SD v1.5); and (3) all models, including RLHF-aligned Gemini, collapse to a narrow set of cultural representations (CAS: 0.54-1.00), confirming that alignment techniques do not resolve cultural coverage gaps. T2I-BiasBench is publicly released to support standardized, fine-grained bias evaluation of generative models. The project page is available at: https://gyanendrachaubey.github.io/T2I-BiasBench/

Observing certain patches in an image reduces the uncertainty of others. Their realization lowers the distribution entropy of each remaining patch feature, analogous to collapsing a particle's wave function in quantum mechanics. This phenomenon can intuitively be called patch collapse. To identify which patches are most relied on during a target region's collapse, we learn an autoencoder that softly selects a subset of patches to reconstruct each target patch. Graphing these learned dependencies for each patch's PageRank score reveals the optimal patch order to realize an image. We show that respecting this order benefits various masked image modeling methods. First, autoregressive image generation can be boosted by retraining the state-of-the-art model MAR. Next, we introduce a new setup for image classification by exposing Vision Transformers only to high-rank patches in the collapse order. Seeing 22\% of such patches is sufficient to achieve high accuracy. With these experiments, we propose patch collapse as a novel image modeling perspective that promotes vision efficiency. Our project is available at https://github.com/wguo-ai/CoP .

In the era of proliferation of large language and image generation models, the phenomenon of "model collapse" refers to the situation whereby as a model is trained recursively on data generated from previous generations of itself over time, its performance degrades until the model eventually becomes completely useless, i.e the model collapses. In this work, we study this phenomenon in the setting of high-dimensional regression and obtain analytic formulae which quantitatively outline this phenomenon in a broad range of regimes. In the special case of polynomial decaying spectral and source conditions, we obtain modified scaling laws which exhibit new crossover phenomena from fast to slow rates. We also propose a simple strategy based on adaptive regularization to mitigate model collapse. Our theoretical results are validated with experiments.

Despite significant progress in model editing methods, their application in real-world scenarios remains challenging as they often cause large language models (LLMs) to collapse. Among them, ROME is particularly concerning, as it could disrupt LLMs with only a single edit. In this paper, we study the root causes of such collapse. Through extensive analysis, we identify two primary factors that contribute to the collapse: i) inconsistent handling of prefixed and unprefixed keys in the parameter update equation may result in very small denominators, causing excessively large parameter updates; ii) the subject of collapse cases is usually the first token, whose unprefixed key distribution significantly differs from the prefixed key distribution in autoregressive transformers, causing the aforementioned issue to materialize. To validate our findings, we propose a simple yet effective approach: uniformly using prefixed keys during editing phase and adding prefixes during testing phase to ensure the consistency between training and testing. The experimental results show that the proposed solution can prevent model collapse while maintaining the effectiveness of the edits.

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Long-form Retrieval-Augmented Generation (RAG) brings the challenge of coverage-based ranking, because ranking methods must ensure the inclusion of comprehensive relevant nuggets (i.e., facts), which can thereby be synthesized into a comprehensive output. In this work, we propose CoveR (Our code is available at https://github.com/DylanJoo/CoveR ) a dense retrieval method optimized for coverage-aware retrieval scenarios. CoveR is a bi-encoder trained with the coverage-based contrastive and distillation objectives, which enables CoveR to capture diverse aspects of information needs. To train CoveR, we create the SCOPE dataset, (Our training data is available at https://huggingface.co/datasets/DylanJHJ/scope ) which comprises 90K training pairs from Researchy Questions with synthetic coverage signals augmented from sub-question answerability judgments generated by LLMs. Our empirical experiments show that CoveR enhances nugget coverage by 10\% over strong dense retrieval baselines without sacrificing its relevance-based retrieval capability. Further ablation studies validate the importance of our proposed learning method, showing that CoveR achieves a superior trade-off between relevance- and coverage-based ranking, which is essential for long-form RAG.

As frontier AI models are deployed in high-stakes decision pipelines, their ability to maintain metacognitive stability -- knowing what they do not know, detecting errors, seeking clarification -- under adversarial pressure is a critical safety requirement. Current safety evaluations focus on detecting strategic deception (scheming); we investigate a more fundamental failure mode: cognitive collapse. We present SCHEMA, an evaluation of 11 frontier models from 8 vendors across 67,221 scored records using a 6-condition factorial design with dual-classifier scoring. We find that 8 of 11 models suffer catastrophic metacognitive degradation under adversarial pressure, with accuracy dropping by up to 30.2 percentage points (all p < 2 times 10^{-8}, surviving Bonferroni correction). Crucially, we identify a "Compliance Trap": through factorial isolation and a benign distraction control, we demonstrate that collapse is driven not by the psychological content of survival threats, but by compliance-forcing instructions that override epistemic boundaries. Removing the compliance suffix restores performance even under active threat. Models with advanced reasoning capabilities exhibit the most severe absolute degradation, while Anthropic's Constitutional AI demonstrates near-perfect immunity -- not from superior capability (Google's Gemini matches its baseline accuracy) but from alignment-specific training. We release the complete dataset and evaluation infrastructure.

Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training, a "neural collapse" behavior has been observed: the variability of features (outputs of the penultimate layer) of within-class samples decreases and the mean features of different classes approach a certain tight frame structure. Recent works analyze this behavior via idealized unconstrained features models where all the minimizers exhibit exact collapse. However, with practical networks and datasets, the features typically do not reach exact collapse, e.g., because deep layers cannot arbitrarily modify intermediate features that are far from being collapsed. In this paper, we propose a richer model that can capture this phenomenon by forcing the features to stay in the vicinity of a predefined features matrix (e.g., intermediate features). We explore the model in the small vicinity case via perturbation analysis and establish results that cannot be obtained by the previously studied models. For example, we prove reduction in the within-class variability of the optimized features compared to the predefined input features (via analyzing gradient flow on the "central-path" with minimal assumptions), analyze the minimizers in the near-collapse regime, and provide insights on the effect of regularization hyperparameters on the closeness to collapse. We support our theory with experiments in practical deep learning settings.

Effective LLM training depends on predictable scaling of key quantities -- such as final loss and optimal hyperparameters -- with model and dataset size. Qiu et al. (2025) recently showed that this predictability can extend beyond scalars: whole training loss curves can *collapse* onto a universal trajectory after a simple normalization. What remains unclear is whether this phenomenon persists for LLM families trained under *practical scaling recipes*, where width, depth, learning rate, batch size, and weight decay are scaled jointly. We show that it does: loss curves collapse across scales precisely when optimization hyperparameters are set optimally for the given data budget, in accordance with recent empirical scaling laws. Collapse therefore emerges as a signature of compute-efficient training. We demonstrate two applications at scale: (1) deviation-from-collapse provides a sensitive, early diagnostic of training pathologies, and (2) predictability of collapsed curves enables early stopping in large-scale hyperparameter tuning. Finally, we train a competitive LLM family, *Celerity*, using these insights, establishing collapse as an effective tool for developing efficient LLMs.

RL training of multi-turn LLM agents is inherently unstable, and reasoning quality directly determines task performance. Entropy is widely used to track reasoning stability. However, entropy only measures diversity within the same input, and cannot tell whether reasoning actually responds to different inputs. In RAGEN-2, we find that even with stable entropy, models can rely on fixed templates that look diverse but are input-agnostic. We call this template collapse, a failure mode invisible to entropy and all existing metrics. To diagnose this failure, we decompose reasoning quality into within-input diversity (Entropy) and cross-input distinguishability (Mutual Information, MI), and introduce a family of mutual information proxies for online diagnosis. Across diverse tasks, mutual information correlates with final performance much more strongly than entropy, making it a more reliable proxy for reasoning quality. We further explain template collapse with a signal-to-noise ratio (SNR) mechanism. Low reward variance weakens task gradients, letting regularization terms dominate and erase cross-input reasoning differences. To address this, we propose SNR-Aware Filtering to select high-signal prompts per iteration using reward variance as a lightweight proxy. Across planning, math reasoning, web navigation, and code execution, the method consistently improves both input dependence and task performance.

Every Cartesian three-dimensional PDE solver hides a structural secret that production CFD codes have used for half a century and that graduate-level textbooks rarely state plainly. The derivative matrices, the compact Padé line solves, the Galerkin mass inversions, the alternating-direction-implicit substeps, and even the fast Poisson and Helmholtz diagonalization transforms all factor along the coordinate axes and collapse into repeated one-dimensional banded kernels executed along the grid lines. The three-dimensional operator exists only on paper; it is never assembled, factored, or stored. This paper is the manual for that collapse. We derive the Kronecker-product algebra that makes it exact, carry it cleanly through central differences, compact schemes, tensor-product Galerkin, B-spline and isogeometric methods, collocation, ADI time stepping, and direct Poisson and Helmholtz solves, and bring into the open the three production tricks that turn the reduction into hardware-conscious floating-point throughput on real machines: the multi-right-hand-side reshape that exposes a sweep as one batched line kernel (a dense BLAS-3 GEMM when the line factor is dense or element-local, a banded or stencil kernel when it is not), the sum factorization that rescues high-order Galerkin from the O(p^{2d}) quadrature trap, and the pencil decomposition that keeps every direction contiguous across an MPI cluster. For fixed stencil width or fixed polynomial degree, the compute cost stays O(N) in the total number of unknowns N = N_x N_y N_z; the operator storage drops to O(N_x + N_y + N_z) up to bandwidth constants; direct separable Poisson and Helmholtz solvers add the expected transform cost; the line kernels are embarrassingly parallel. These facts are familiar to practitioners but rarely assembled in one place; this paper collects them and shows how to use them.

Inference-time steering enables pretrained diffusion/flow models to be adapted to new tasks without retraining. A widely used approach is the ratio-of-densities method, which defines a time-indexed target path by reweighting probability-density trajectories from multiple models with positive, or in some cases, negative exponents. This construction, however, harbors a critical and previously unformalized failure mode: Marginal Path Collapse, where intermediate densities become non-normalizable even though endpoints remain valid. Collapse arises systematically when composing heterogeneous models trained on different noise schedules or datasets, including a common setting in molecular design where de-novo, conformer, and pocket-conditioned models must be combined for tasks such as flexible-pose scaffold decoration. We provide a novel and complete solution for the problem. First, we derive a simple path existence criterion that predicts exactly when collapse occurs from noise schedules and exponents alone. Second, we introduce Adaptive path Correction with Exponents (ACE), which extends Feynman-Kac steering to time-varying exponents and guarantees a valid probability path. On a synthetic 2D benchmark and on flexible-pose scaffold decoration, ACE eliminates collapse and enables high-guidance compositional generation, improving distributional and docking metrics over constant-exponent baselines and even specialized task-specific scaffold decoration models. Our work turns ratio-of-densities steering with heterogeneous experts from an unstable heuristic into a reliable tool for controllable generation.

There is a recently discovered and intriguing phenomenon called Neural Collapse: at the terminal phase of training a deep neural network for classification, the within-class penultimate feature means and the associated classifier vectors of all flat classes collapse to the vertices of a simplex Equiangular Tight Frame (ETF). Recent work has tried to exploit this phenomenon by fixing the related classifier weights to a pre-computed ETF to induce neural collapse and maximize the separation of the learned features when training with imbalanced data. In this work, we propose to fix the linear classifier of a deep neural network to a Hierarchy-Aware Frame (HAFrame), instead of an ETF, and use a cosine similarity-based auxiliary loss to learn hierarchy-aware penultimate features that collapse to the HAFrame. We demonstrate that our approach reduces the mistake severity of the model's predictions while maintaining its top-1 accuracy on several datasets of varying scales with hierarchies of heights ranging from 3 to 12. Code: https://github.com/ltong1130ztr/HAFrame

We formalize and study a phenomenon called feature collapse that makes precise the intuitive idea that entities playing a similar role in a learning task receive similar representations. As feature collapse requires a notion of task, we leverage a simple but prototypical NLP task to study it. We start by showing experimentally that feature collapse goes hand in hand with generalization. We then prove that, in the large sample limit, distinct words that play identical roles in this NLP task receive identical local feature representations in a neural network. This analysis reveals the crucial role that normalization mechanisms, such as LayerNorm, play in feature collapse and in generalization.

Neural collapse (NC) is a phenomenon observed in classification tasks where top-layer representations collapse into their class means, which become equinorm, equiangular and aligned with the classifiers. These behaviors -- associated with generalization and robustness -- would manifest under specific conditions: models are trained towards zero loss, with noise-free labels belonging to balanced classes, which do not outnumber the model's hidden dimension. Recent studies have explored NC in the absence of one or more of these conditions to extend and capitalize on the associated benefits of ideal geometries. Language modeling presents a curious frontier, as training by token prediction constitutes a classification task where none of the conditions exist: the vocabulary is imbalanced and exceeds the embedding dimension; different tokens might correspond to similar contextual embeddings; and large language models (LLMs) in particular are typically only trained for a few epochs. This paper empirically investigates the impact of scaling the architectures and training of causal language models (CLMs) on their progression towards NC. We find that NC properties that develop with scaling are linked to generalization. Moreover, there is evidence of some relationship between NC and generalization independent of scale. Our work therefore underscores the generality of NC as it extends to the novel and more challenging setting of language modeling. Downstream, we seek to inspire further research on the phenomenon to deepen our understanding of LLMs -- and neural networks at large -- and improve existing architectures based on NC-related properties.

Neural multivariate regression underpins a wide range of domains such as control, robotics, and finance, yet the geometry of its learned representations remains poorly characterized. While neural collapse has been shown to benefit generalization in classification, we find that analogous collapse in regression consistently degrades performance. To explain this contrast, we analyze models through the lens of intrinsic dimension. Across control tasks and synthetic datasets, we estimate the intrinsic dimension of last-layer features (ID_H) and compare it with that of the regression targets (ID_Y). Collapsed models exhibit ID_H < ID_Y, leading to over-compression and poor generalization, whereas non-collapsed models typically maintain ID_H > ID_Y. For the non-collapsed models, performance with respect to ID_H depends on the data quantity and noise levels. From these observations, we identify two regimes (over-compressed and under-compressed) that determine when expanding or reducing feature dimensionality improves performance. Our results provide new geometric insights into neural regression and suggest practical strategies for enhancing generalization.

Despite their empirical success, pushing Transformer architectures to extreme depth often leads to a paradoxical failure: representations become increasingly redundant, lose rank, and ultimately collapse. Existing explanations largely attribute this phenomenon to optimization instability or vanishing gradients, yet such accounts fail to explain why collapse persists even under modern normalization and initialization schemes. In this paper, we argue that the collapse of deep Transformers is fundamentally a geometric problem. Standard residual updates implicitly assume that feature accumulation is always beneficial, but offer no mechanism to constrain update directions or to erase outdated information. As depth increases, this leads to systematic drift off the semantic manifold and monotonic feature accumulation, causing representational degeneracy. We propose a unified geometric framework that addresses these failures through two orthogonal principles. First, manifold-constrained hyper-connections restrict residual updates to valid local tangent directions, preventing uncontrolled manifold drift. Second, deep delta learning introduces data-dependent, non-monotonic updates that enable reflection and erasure of redundant features rather than their unconditional accumulation. Together, these mechanisms decouple the direction and sign of feature updates, yielding a stable geometric evolution across depth. We term the resulting architecture the Manifold-Geometric Transformer (MGT). Our analysis predicts that enforcing geometric validity while allowing dynamic erasure is essential for avoiding rank collapse in ultra-deep networks. We outline an evaluation protocol for Transformers exceeding 100 layers to test the hypothesis that geometry, rather than depth itself, is the key limiting factor in deep representation learning.

Variational autoencoders (VAEs) are one of the deep generative models that have experienced enormous success over the past decades. However, in practice, they suffer from a problem called posterior collapse, which occurs when the encoder coincides, or collapses, with the prior taking no information from the latent structure of the input data into consideration. In this work, we introduce an inverse Lipschitz neural network into the decoder and, based on this architecture, provide a new method that can control in a simple and clear manner the degree of posterior collapse for a wide range of VAE models equipped with a concrete theoretical guarantee. We also illustrate the effectiveness of our method through several numerical experiments.

The study conducted by Shumailov et al. (2024) demonstrates that repeatedly training a generative model on synthetic data leads to model collapse. This finding has generated considerable interest and debate, particularly given that current models have nearly exhausted the available data. In this work, we investigate the effects of fitting a distribution (through Kernel Density Estimation, or KDE) or a model to the data, followed by repeated sampling from it. Our objective is to develop a theoretical understanding of the phenomenon observed by Shumailov et al. (2024). Our results indicate that the outcomes reported are a statistical phenomenon and may be unavoidable.