Model Weights

The emergence of low-dimensional structures in the spectra of neural network weight matrices is a common empirical feature of trained models, but the dynamical origin of this phenomenon during learning remains an open problem. We formulate neural network training as the stochastic evolution of an initially random matrix ensemble, driven by stochastic gradient descent (SGD) updates that reshape the spectral bulk while amplifying signal strength. This induces a Baik-Ben Arous-Péché (BBP) transition during training, where isolated eigenvalues detach from the random bulk distribution, providing a dynamical framework for representation formation in high-dimensional learning dynamics. We demonstrate this in a solvable linear teacher-student model, where spectral evolution is analytically tractable and a phase diagram of trainability governed by the step size (or learning rate) and initial weight variance is obtained, and subsequently extend our formalism beyond the linear regime to nonlinear and stochastic settings. Numerical simulations in realistic settings support this picture, showing robust emergence of spectral alignment during training. Our results suggest that spectral analysis may provide a unified perspective of stochastic learning dynamics, linking trainability, optimisation hyperparameters, spectral phase transitions, and representation learning in neural networks.

Classical universal approximation theorems establish the expressive power of sigmoidal multilayer perceptrons, but they do not prescribe how initial weights should encode the geometry of a data distribution. We propose S-GAI, a spectral geometry-aware initialization framework for one-hidden-layer sigmoidal MLPs. Starting from the constructive idea that sigmoid units can act as smooth half-space gates, we move from hand-specified planar geometry to class-wise spectral geometry estimated from image data. For each class, SVD provides a mean, principal directions, and spectral scales. An energy threshold selects the retained directions, and each retained direction is represented by two sigmoid gates. These class-specific gates form a shared hidden layer initialized directly from the training set. We also formulate a SVD-based subspace classifier as a non-neural geometric reference, which tests whether the estimated spectral class geometry is already discriminative before being embedded into the MLP. Experiments on MNIST, Fashion-MNIST, and a more challenging CIFAR-10 test show that the S-GAI-initialized MLP starts from a substantially more informative hidden state than Xavier initialization and reaches comparable final accuracy under full training. When the hidden layer is frozen, training only the output layer still gives stronger performance than frozen random gates, providing evidence that S-GAI effectively embeds class-wise spectral geometry into the MLP.

Adapting a foundation vision-language encoder to a specialized retrieval task creates a fundamental tradeoff: gains on the target distribution come at the cost of the foundation model's broad generalization, and fashion retrieval is a stringent instance of this problem. We present ZooClaw-FashionSigLIP2, a fashion-specialized SigLIP2-base model that resolves this tradeoff with a simple recipe -- full fine-tuning with knowledge distillation on curated in-domain data, followed by \wiseft~\citep{wortsman2022wiseft} weight interpolation with the base model -- and outperforms LoRA, larger backbones (up to 1B parameters), and external training data. Under fair evaluation, ZooClaw-FashionSigLIP2 outperforms all baselines on every benchmark in our suite. In addition, we release ZooClaw-Fashion, a new high-quality fashion retrieval benchmark, and a systematic quality analysis of widely-used benchmarks that exposes and mitigates structural biases in their public ground truth. We open-source the model weights and all evaluation artifacts to facilitate future research.

Muon-type optimizers construct update directions for dense neural-network weights by applying a finite Newton-Schulz map to momentum-gradient matrices. For an $H \times W$ matrix, with $r=\min\{H,W\}$ and $s=\max\{H,W\}$, $K$ steps of the full-matrix Newton-Schulz update require $O(r^2 s K)$ work and couple all rows and columns through repeated Gram matrix products. We introduce Hierarchical Muon (HiMuon), a tiled Newton-Schulz scheme for Muon-type optimization. HiMuon partitions each momentum-gradient matrix into $T \times T$ tiles, applies the same finite Newton-Schulz map independently to each tile, and reassembles the results. For finite $T$ below the matrix dimensions, HiMuon defines a local matrix-function map rather than a convergent approximation to the full-matrix update: spectral interactions are preserved within tiles and discarded across tile boundaries. For fixed finite $T$, the leading Newton-Schulz work decreases to $O(H W T K)$, and the computation decomposes into independent small dense matrix operations. This structure enables tile-size-dependent GPU kernels, cross-layer batching, memory-bounded chunking, and runtime tile-size schedules. Experiments on transformer training and controlled matrix-function diagnostics show that HiMuon improves optimizer-step efficiency while keeping training behavior close to full-matrix Muon in the tested regimes.

Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object. Yet every weight matrix carries two distinct quantities -- a \emph{magnitude} and a \emph{direction} -- and all optimizers stepping in the matrix as a whole couple their dynamics: the directional change from an update depends on the current magnitude, while the magnitude drifts as a byproduct of learning the direction, so neither is governed directly by the learning rate. Typical training therefore leans on surrounding recipes such as weight decay and warmup to keep learning stable at scale, though these regulate the coupling only indirectly; other recent methods instead constrain the weight to a fixed-norm sphere, but add no learnable magnitude, leaving scale control to normalization layers alone. We propose \emph{Magnitude--Direction (MD) Decoupling}, an optimizer modification that factorizes each weight into a fixed-norm direction on a hypersphere and learnable per-row and per-column magnitude gains, updated at separate learning rates, all while the model still sees a single fused weight tensor. The method is agnostic to the base optimizer and removes the need for weight decay and warmup. Across both Adam and Muon, MD Decoupling improves on well-tuned baselines, transfers the optimal LR across model width without retuning, and continues to help at scale on large Mixture-of-Experts (MoE) models. Treating magnitude and direction as separately controlled quantities thus yields more predictable training dynamics and a simple, broadly applicable improvement to modern optimizers.

Random initialization of convolutional filters does not use the training images. Previous work has shown that image patches can be copied into the first layer, and that k-means or principal components of patches can serve as filters. This paper compares four initializations of the first layer of a small convolutional network, with every other factor held fixed: He initialization, random mean-centered patches, principal components of those patches, and k-means centroids. Pre-Warm, our proposed methodology, is the rule-based use of both dictionaries: the patch count follows the filter count and a foreground density, both dictionaries are built from a single minibatch, and whichever of principal components or k-means better reconstructs those patches is written into the first half of the filter bank, rather than chosen by a validation search. The remaining filters stay random. On five datasets, principal components improve CIFAR-10 and CIFAR-100 relative to He initialization, and k-means improves SVHN and MNIST, and is the stronger of the two on Fashion-MNIST; copying raw patches does not reproduce those color-set gains. Use principal components on photographic patches and k-means on stroke-like patches; the first-batch reconstruction check recovers that split.

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

Matrix-aware optimizers such as Muon and Muown have recently shown strong empirical performance for pre-training Transformers. In particular, Muown separates each weight matrix into row magnitudes and an un-normalized direction variable, updating the former with Adam and the latter with Muon. We show that the directional update of Muown is equivalent to a Riemannian step on the normalized directions, while the magnitude of the un-normalized parameterization only modulates the angular step size. This explains the step-size stability of Muown and suggests making the angular step size explicit. The resulting method, AngularMuown, optimizes directly over the normalized directions and uses a schedulable angular multiplier decoupled from the radial magnitude update. AngularMuown improves over Muown and, at the time of writing, a preliminary version is leading the per-optimizer category of the modded nanoGPT speedrunning competition. Further experiments on Qwen2-0.5B, and 1.1B parameter mixture-of-experts models confirm the algorithm scales beyond small models. An implementation of the algorithm is available at https://github.com/fhueb/angular-muown

Linear mode connectivity (LMC) provides a promising foundation for understanding and merging independently trained neural networks, but existing methods typically optimize the interpolation path from only one model endpoint, limiting their scalability and effectiveness for large pretrained transformers. We propose a novel and scalable framework for enabling LMC-based model merging to {\em billion-parameter pretrained transformers}. Our method applies properly parameterized functionality-preserving weight transformations to align functionally equivalent solutions, and introduces a dual learning procedure in which both models jointly learn their corresponding transformations toward a shared linear interpolation path. This bidirectional optimization substantially reduces interpolation barriers and enables more reliable merging across large-scale architectures. Empirically, we show that our approach achieves near-zero loss barriers on WikiText for language models with medium-sized parameters, representing, to our knowledge, the first demonstration of near-barrier-free linear connectivity at this scale. In the vision domain, ViT-L maintains above 69% ImageNet top-1 accuracy throughout the interpolation path, while modern billion-parameter LLMs exhibit only small loss barriers. These results suggest that properly resolving parameter symmetries enables large pretrained Transformers to be connected and merged through simple linear paths with substantially improved interpolation performance. Code: https://github.com/VILA-Lab/Dual-Learned-Matching .

The integration of multiple thematic data layers into a single composite map, known as the cartographic synthesis problem, is typically addressed through expert-driven weighting schemes. This study presents a multi-objective formulation of cartographic synthesis grounded in spatial autocorrelation structure. We develop a bi-objective evolutionary framework, GIS-moGA, that estimates layer weights by simultaneously maximizing global spatial structure, measured by Global Moran's I, and minimizing local spatial heterogeneity, measured by the variance of Local Indicators of Spatial Association (LISA). Because naive evaluation of spatial relationships requires O(N^2) operations, direct computation becomes impractical for larger datasets. We address this challenge by exploiting the 97.7% sparsity of queen contiguity matrices, reducing effective complexity to O(N k) and enabling scalable municipal-level analysis. The framework is evaluated on a high-dimensional spatial epidemiology dataset with N = 523 units from Araraquara, Brazil. A 64-scenario experimental design is used to examine evolutionary behavior across parameter settings. Results show that higher mutation rates are important for maintaining population diversity and preventing premature convergence in spatially autocorrelated fitness landscapes, where crossover operators can disrupt geographically coherent structures. Compared with expert-derived Analytic Hierarchy Process baselines, the resulting Pareto fronts show substantial hypervolume gains and significant improvements in spatial coherence (p < 0.001, Cliff's delta = 0.87). These findings provide a systematic and scalable framework for data-driven geographic multi-criteria decision analysis.

In recent years, weight quantization that encodes the learnable parameters of large language models in an $n$-bit format has garnered significant attention due to its potential for model compression and inference acceleration. Many practical techniques have been developed; however, the theoretical understanding of many aspects, especially the approximation and degradation of expressive power as the number of quantization bits decreases, remains unclear. In this paper, we provide a theoretical investigation into the expressive capability of large language models relative to the number of quantization bits. We argue that 1.58-bit is the limiting precision for weight quantization by establishing the universal approximation and expressive collapse properties of weight-quantized models with respect to the number of quantization bits. Additionally, we confirm that weight quantization leads to expressive degradation, in which the expressive capacity of weight-quantized models degrades polynomially as the number of quantization bits decreases. These theoretical findings provide a solid foundation for advancing weight quantization in the context of scaling laws and shed insights for future research in model compression and inference acceleration.

Large multi-modal language models are increasingly deployed in high-stakes domains, making well-calibrated uncertainty essential. Traditional Bayesian methods approximate posteriors over all model weights, which becomes intractable for modern large models. For this reason, recent work instead considers Bayesian low-rank adaptation to enable tractable posterior approximation. Due to a lack of a standardized benchmark to evaluate these approaches, it remains unclear where these methods provide meaningful benefits. To fill this gap, we introduce Bayesian Adaptation Gym (BAG), a benchmark for the Bayesian adaptation of multi-modal language models. BAG provides reference implementations of classic Bayesian baselines and state-of-the-art adaptation methods, along with a multi-modal dataset and task suite designed to probe calibration, robustness under distribution shift, and decision-making under uncertainty via active learning. Using BAG, we conduct and report extensive experiments across model sizes, datasets, and tasks to highlight the successes and failures of current Bayesian adaptation approaches. To enable further research, BAG is fully open source: https://github.com/SRI-CSL/BayesAdapt.

Open-weight Large Language Models (LLMs) enable scientific progress and broad deployment. However, they make it difficult to control access to sensitive capabilities. Current practice either suppresses dangerous capabilities before release or mediates access through closed services that use specialized model variants, input/output monitors, and API permissions. The former is susceptible to jailbreaks while sacrificing capability for all users to mitigate the risks posed by a few, and the latter is fundamentally incompatible with open-weight release. In this paper, we propose Tiered Language Models (TLMs), where a single set of released weights supports multiple capability levels. In its default public configuration, a TLM behaves as a conventional LLM. A compact secret key specifies a permutation over a small parameter subset, inducing an alternative computation graph over the same weights that exposes additional capabilities. We develop a training protocol that jointly pretrains both configurations from scratch, then fine-tunes the keyed configuration on private data with regularization to preserve the public model's behavior. We pretrain 180M- and 650M-parameter TLMs and demonstrate that the keyed configuration can acquire a new language, gain instruction-following ability, and memorize private factual knowledge, whereas the public configuration exhibits none of these capabilities. Moreover, we show that our approach extends naturally to multiple hierarchical tiers. Because authorization operates on the model's weight structure rather than in the input space, the mechanism resists fine-tuning-based extraction and partial key compromise. In general, TLMs take a step toward reconciling open-weight release with selective capability control.

Reinforcement learning (RL) post-training of Diffusion Transformers (DiTs) is prohibitively expensive, requiring thousands of high-end GPUs. Existing works explore two directions to reduce cost: seed exploration improves training convergence by selecting high-contrast samples, yet adds compute to the critical path; spot GPUs offer 69--77% lower cost, yet sit idle during training because DiT rollouts finish nearly simultaneously, which prevents LLM-style pipelining of rollout with training. Spot preemptions further break Sequence Parallelism (SP) groups, fragmenting GPU topology. We present Spotlight, the first system that harvests spot GPUs for DiT RL post-training. Spotlight rests on two key insights we devise: (1)~we show that exploration can tolerate stale model weights because exploration that uses the model weights from the previous iteration preserves the relative ranking of random seeds, allowing exploration to run on idle spot GPUs during training. (2)~SP reconfiguration can reuse on-node state, reducing group recovery from minutes to sub-second launches. Built on these insights, Spotlight introduces three techniques: a bandit-based exploration planner that maximizes reward variance within the training time budget, elastic sequence parallelism that reconfigures SP groups on the fly via persistent schedulers and intra-node weight copying, and a preemption-aware pull-based request scheduler that balances load and commits in-flight state upon preemption. We implement Spotlight on the open-source RL platform ROLL and evaluate it on Qwen-Image post-training. Spotlight reaches the same target validation score $4\times$ faster than baselines, reducing total cost by $1.4$-$6.4\times$ while achieving superior image quality on DeepSeek-OCR and Geneval datasets with resolution $512\times512$ and $1280\times1280$.

Grokking, the delayed jump from memorization to generalization, is usually tied to the weight norm: a smaller norm generalizes sooner. We ask what the norm actually controls. Holding the weight norm fixed by clamping and varying only an output temperature, we slide the grokking delay across its entire norm-induced range under cross-entropy; matching the effective logit scale back to baseline recovers about 85% of the delay at two moduli. Across a grid of norms and temperatures the delay collapses onto the logit scale alone (R2 = 0.97), with the norm adding 1-2% beyond it. The effect is loss-dependent: under mean-squared error the logit scale is pinned and the norm acts through a different route. A memorization control, a float64 softmax-collapse audit, and a no-LayerNorm transformer point to the same channel. Forking arms from one identical state, the delay follows the held norm value and not the clamp operation, which closes a rescaling-artifact concern. The proximal variable is the logit scale and the softmax saturation it drives; the weight norm is only an upstream handle. All numbers, tables, and figures reproduce from released code and data.

Matrix based optimizers such as Muon can substantially speed up language model pretraining, but their gains over AdamW are observed to shrink as model size and data scale grow when using standard constant decoupled weight decay. We propose Hyperball, a simple optimizer wrapper that addresses this issue. Given a base optimizer such as Adam or Muon, Hyperball sets the Frobenius norms of weight matrices and their corresponding optimizer updates to fixed constants. On Qwen3 style models up to 1.2B parameters, Muon Hyperball achieves 20--30% token equivalent speedup over weight decay baselines. Hyperball also improves learning rate transfer across widths and depths compared to decoupled weight decay. This method is motivated by prior theory showing that training with weight decay leads to an equilibrium weight norm that only depends on the training hyperparameters. Through this mechanism, the weight decay then decides the angular learning rate, i.e. how fast the direction of the weight matrix changes.

Federated Learning is rapidly evolving beyond the exchange of traditional model weights and gradients, yet existing definitions fail to capture the full scope of modern payloads like synthetic data and federated analytics. This paper addresses the gap by proposing a formal mathematical definition of a federated message that accounts for both utility and privacy. We introduce a taxonomy that organizes these exchanges into three categories: model structures, statistical summaries, and data-conditioned representations. By evaluating these groups based on computational demands, communication costs, and privacy risks, we provide a clearer understanding of the trade-offs involved in decentralized training. Our review of 202 recent publications highlights a significant shift since 2021 toward diverse messaging paradigms, signaling a move away from standard deep learning updates toward more specialized information sharing. This framework provides a structured path for future research to optimize federated systems for varying hardware and security requirements.

Large-scale neural network training increasingly relies on matrix-aware optimizers that exploit the structure of weight parameters beyond element-wise adaptation. However, existing matrix-aware methods such as Muon have an underappreciated vulnerability: their core operation, Newton-Schulz iteration, depends critically on input conditioning, yet the raw momentum matrices exhibit severe coordinate-wise scale heterogeneity. In this paper, we first verify this scale heterogeneity through a chi-square uniformity test, showing that intra-matrix scale imbalance is prevalent across Transformer layers and that coordinate whitening effectively corrects it. Motivated by this finding, we propose Zeta, a dual whitening optimizer that applies coordinate whitening and spectral whitening in a strictly ordered pipeline. The ordering is not a tunable choice but follows from a mathematical dependency: coordinate whitening establishes the statistical isotropy that spectral whitening requires to function reliably. We further prove that this dual pipeline strictly reduces orthogonalization error relative to pure spectral methods by improving the condition number of the input. Empirically, Zeta matches or surpasses strong baselines across language modeling (0.6B to 8B parameters), mixture-of-experts architectures, and vision tasks, demonstrating that resolving scale imbalance before orthogonalization leads to faster convergence and better generalization. Code is available at https://github.com/AIGCodeOS/aigcode_zeta_optimizer.

Building on a two-parameter Weibull framework for diagnosing transformer weight distributions, we study why the Weibull weight-scale parameter $λ$ grows, overshoots, and then relaxes during AdamW training. We derive a leading-order three-force decomposition of the squared weight norm from the AdamW update: an alignment force measuring the correlation between weights and the adaptive update direction, an injection force from adaptive step magnitude, and a decay force from decoupled weight decay. On self-trained Pythia-70M models with ground-truth optimizer moments, alignment dominates the rise phase, contributing 88-94% of the absolute force budget across four random seeds and remaining robust to super-weight removal. Near saturation, alignment and decay approach balance, explaining the transition from weight-scale growth to relaxation. These force dynamics directly govern the squared-norm component underlying $λ(t)$; the remaining RMS-to-Weibull reconstruction offset is measurable and decomposes into bridge and integration components, totaling approximately 5-6% in densely sampled regions. To extend the analysis to real models where optimizer moments are unavailable, we introduce a spline displacement method that recovers the alignment force from sparse checkpoints with approximately 92-94% accuracy, about twice the naive two-point baseline. We further observe that the peak value of $λ(t)$ varies with training-data coherence in our experiments, suggesting a data-dependent component of weight-scale growth that we leave to a controlled follow-up study. Code and data are available at https://github.com/tiexinding/NPM-Weibull-public.

We propose a preconditioning (PC) layer, a weight parameterization via polynomial preconditioner that ensures stable weight conditioning throughout LLM training. The PC module reshapes the singular-value spectrum of weight matrices via low-degree polynomial preconditioning. After training, the preconditioned weights can be merged back into the original architecture, incurring no inference overhead. We demonstrate the advantage of the proposed PC layer over standard transformers in Llama-1B pre-training, for both the AdamW and Muon optimizers. Theoretically, we justify this spectrum-control principle by proving that uniformly bounding each layer's singular values ensures geometric convergence of gradient descent to global minima, for certain deep linear networks. Our code is available at https://github.com/Empath-aln/PC-layer.

Large-scale dynamic weighted directed networks (DWDNs) are widely used to model time-varying interactions among nodes. Latent factorization of tensors (LFT) extracts target knowledge from DWDNs via low-rank embedding. However, similar to many machine learning models, the performance of LFT heavily depends on the selection of hyperparameters. In practice, these parameters are often tuned manually or through grid search, which requires significant computational resources and human effort. Motivated by this challenge, this paper proposes an automated hyperparameter optimization framework based on Differential Evolution (DE) for LFT (DE-LFT). The proposed method integrates DE into the training process of the LFT model to automatically learn optimal regularization parameters $λ_1$, $λ_2$ and $λ_3$. As a result, the model can adaptively search the hyperparameter space and improve prediction accuracy. Experimental results on four real-world datasets demonstrate that the proposed approach achieves lower MAE and RMSE compared with manually tuned baselines while reducing the need for extensive parameter tuning.

Aggressive weight quantization to 2-bit precision offers substantial throughput and memory gains for large language model (LLM) inference, but typically incurs severe accuracy degradation. These gains are particularly relevant for edge and on-device deployment, where memory capacity and bandwidth are primary constraints. In this work, we extend Recover-LoRA -- a lightweight, data-free accuracy recovery method originally developed for general model weight corruption -- to the setting of ultra-low-bit quantization. We propose a selective mixed-precision strategy in which only gate and up projection layers of the MLP are quantized to 2-bit (W2), while all other linear layers remain at higher precision, yielding a mixed-precision GateUp configuration. We demonstrate via roofline analysis across three model families (4B--20B) and two hardware platforms that a W4/W2-GateUp deployment (4-bit base with 2-bit gate/up) delivers 7.5--23.3% TPS improvement over uniform W4 depending on model and context length, while confining quantization error to a predictable subset of layers. We then apply Recover-LoRA -- training low-rank adapters on the quantized layers via logit distillation with synthetic data -- to recover accuracy lost from 2-bit quantization of the gate and up layers. In a case study on Qwen3-4B, Recover-LoRA achieves 80--95% accuracy recovery on 9 of 12 benchmarks, using only 10k synthetic training samples and no labeled data. We further demonstrate that synthetic data performs comparably to curated labeled data for distillation-based recovery, and that recovery generalizes to out-of-distribution evaluation tasks. Our results present Recover-LoRA as a practical post-quantization accuracy recovery tool for aggressive weight compression in deployment settings.

Orthonormalized update rules have rapidly become a leading choice of optimizer for training large language models, with recent open-source state-of-the-art models adopting Muon. To keep these updates tractable, Muon performs the orthonormalization with the Newton--Schulz (NS) iteration. Since NS is only approximate, directions with small singular values fail to be orthonormalized. In Muon, NS is applied to the momentum matrix at every step, yet little is known about how the singular value spectrum of these momentum matrices behaves during training, or how that behavior changes with model size. We present the first systematic study of this question. Tracking singular value quantiles of the momentum buffer across layers in models ranging from 77M to 2.8B parameters, we observe a consistent picture: after a short burn-in, the quantiles stabilize at a value determined by the layer type and model size. These stabilization values follow remarkably clean power laws in model size, with layer-dependent exponents. Layers up to mid-late depth scale very mildly with model size $M$ (around $M^{-0.25}$), so the standard 5-step NS configuration used at academic scale will continue to orthonormalize them at much larger scales. Some of the late layers, however, scale much more aggressively (up to $M^{-0.96}$) and will fall into the NS failure regime at frontier scale unless one uses more NS iterations or better-tuned coefficients. NS iterations are computationally expensive at scale; our laws give practitioners a principled, layer-aware recipe for choosing the minimum NS configuration that still orthonormalizes the directions that matter -- avoiding unnecessary computation without sacrificing update quality.

Two-bit weight quantization is attractive for memory-efficient LLM inference, but the standard W2 level set {-2,-1,0,+1} often collapses under aggressive W2A4/KV4 settings. We study the scalar level-set geometry of two-bit weights in a Hadamard-rotated quantization pipeline. Conventional asymmetric W2 substantially improves over the standard level set, indicating that W2A4 failure is not only a bit-width problem but also a reconstruction-level problem. Across all 224 linear modules in each of LLaMA-2-7B and LLaMA-3.1-8B, pretrained weights are already nearly zero-centered, while Hadamard rotation primarily Gaussianizes their standardized shape: excess kurtosis and Q-Q error drop by orders of magnitude. Based on this approximate zero-centered Gaussian-like source model, we propose Qift, a fixed no-zero W2 level set for rotated W2A4/KV4 inference. The main level set is {+/-0.5, +/-1.5}, equivalently {+/-1, +/-3} under a half-scale reparameterization; a power-of-two variant uses {+/-1, +/-4} for sign-and-shift decoded weight application. Qift redesigns the fixed two-bit code-to-level mapping and is training-free, learned-codebook-free, group-grid-free, and zero-point-free, retaining the standard per-channel scale. A scale-invariant ratio analysis identifies an effective inner/outer centroid ratio range of 0.25 to 0.33, explaining why mirror no-zero (MNZ), Lloyd, NF2, and PoT-MNZ perform well while {+/-1, +/-2} does not. On both models, the no-zero level sets consistently improve pure W2A4 perplexity, L-layer mixed W2/W4 perplexity, downstream accuracy, and GPTQ residual behavior over the standard W2 level set. At L=16 mixed precision, they substantially narrow the gap to W3A4 while keeping half of the transformer layers at two-bit precision, giving a simple, source-aware, and deployment-friendly alternative to more complex learned W2 codebooks.

Sparse Mixture of Experts (SMoE) architectures improve the training efficiency of Large Language Models (LLMs) by routing input tokens to a selected subset of specialized experts. Despite their remarkable success, both training and inference in SMoE models suffer from the expert collapse issue (Chi et al., 2022), which degrades model performance. Prior studies primarily focus on improving the router; however, such methods rely on training from scratch or fine-tuning, which requires high computational and data-processing costs. Furthermore, we demonstrate that, despite these efforts, the issue persists when advancing well-pretrained SMoE models, as evidenced by both theoretical and empirical results. To fill that gap, we analyze the advanced SMoE models and observe that the eigenvectors of expert weight matrices encode rich semantic information, pointing to an effective alternative to conventional routing strategies. Building on this insight, we propose Singular Value Decomposition SMoE (SSMoE), a novel and training-free framework that leverages spectral properties of the expert weights to address the collapse issue and enhance model performance. Extensive experiments across diverse language and vision tasks, under both clean and corrupt data settings, demonstrate the strong generalization and robustness of SSMoE. Our findings highlight how a deeper understanding of model internals can guide the development of more effective SMoE architectures. Our implementation is publicly available at https://github.com/giangdip2410/SSMoE.

Recent SVD based compression methods for large language models like SVD LLM and Basis Sharing can be unified under one optimization problem. While mathematical proofs and tests on Pythia models show this unified approach improves weight reconstruction error by up to 46% percent it fails in practical tasks. Downstream metrics like perplexity and accuracy severely degrade compared to standard per layer SVD LLM. The authors explain this failure mechanistically. Although the bundle method mathematically couples adjacent layers the transformer residual stream actually decouples them during forward passes. Thus per layer optimality matters more than joint cross layer optimization. The paper concludes that weight space reconstruction is a flawed objective for cross layer compression and future methods must focus on per layer activation reconstruction instead.

Various algorithms have been proposed to address the challenges posed by class-imbalanced learning from real-world data with long-tailed distributions. While these algorithms reduce prediction bias through rebalancing techniques, they often introduce increased prediction variance as a trade-off. Several multi-expert learning algorithms aim to address this variance but involve complex procedures. We propose a new multi-expert learning algorithm, called the dual-axis multi-expert learning (DAMEL), which reduces both bias and variance of predictions by using multiple experts along both representation and time axes. Along the representation axis, DAMEL concatenates the representations of multiple experts and trains an auxiliary balanced classifier simultaneously with the concatenated representations. Along the time axis, DAMEL aggregates network weights across training epochs, employing these aggregated weights during testing. Experimental results demonstrate that DAMEL reduces both bias and variance of predictions, highlighting its effectiveness in class-imbalanced learning.

Deep networks are powerful function approximators, but they typically store many different computations in shared weight matrices, making it difficult to selectively reuse or adapt parts of them when a familiar structure appears in novel combinations. We introduce the Vector Network (VN), a hierarchical recurrent architecture in which each layer replaces a fixed weight matrix with a library of reusable rank-1 weight atoms. For each input, VN minimizes a layer-local energy to infer a sparse set of active weight atoms and their coefficients, jointly constrained by bottom-up input reconstruction and top-down feedback consistency. These weight atom coefficients then compose an input-specific low-rank weight matrix for that sample. After convergence, slow learning updates only the selected weight atoms through local residual signals scaled by the inferred coefficients. We evaluate VN on four compositional benchmarks spanning 1D signals, 2D spatial decoding, N-body dynamics, and compositional MNIST. VN matches strong baselines in distribution while often achieving out-of-distribution error about an order of magnitude lower when familiar factors must be recombined in novel ways. Vector networks thus make compositional generalization a structural property of the architecture and inference process rather than a brittle byproduct of fitting many behaviors into one shared dense parameter substrate.

Normalization layers in modern large language models (LLMs) consist of a deterministic normalization operation and a learnable scale vector. While the normalization operation has been extensively studied, the scale vector remains poorly understood despite its ubiquitous use. In this work, we present a systematic study of scale vectors in LLMs from the perspectives of expressivity, optimization, and architectural structure. First, we show empirically that although scale vectors constitute only a negligible fraction of model parameters, removing them substantially degrades LLM pre-training. Our theory further shows that, in Pre-Norm architectures, scale vectors do not increase expressivity; instead, they improve optimization through a self-amplifying preconditioning effect on subsequent linear mappings. Second, we investigate the role of weight decay for scale vectors. By distinguishing Input-Norm and Output-Norm layers, we theoretically show that weight decay is beneficial for the former but harmful for the latter, due to their distinct roles in optimization and expressivity. Third, motivated by this understanding, we propose three lightweight and complementary improvements to scale vectors: branch-specific heterogeneity, improved placement around linear mappings, and magnitude-direction reparameterization. Both theory and experiments show that each improvement yields consistent gains. Finally, we combine these improvements into a unified scale-vector strategy and evaluate it through extensive LLM pre-training experiments on dense and mixture-of-experts models ranging from 0.12B to 2B parameters, across multiple optimizers and learning rate schedules, under industrial-scale token budgets. The unified strategy consistently achieves lower terminal loss than well-tuned baselines and exhibits more favorable scaling behavior, while adding negligible parameter and computational overhead.

Scalar post-training quantizers discard pairwise coordinate structure within weight rows. We introduce QAM-W (Quadrature Amplitude Modulation for Weights), a codec that recovers this structure: each row is L2-normalized, block-Hadamard rotated, paired into 2D coordinates, and quantized against a single Lloyd-Max codebook trained on the unit circular Gaussian, with activation-aware per-channel scaling. In a cross-model study spanning five LLMs from four families (1.1B--13B parameters) and eight quantized configurations, the activation-aware variant at $\approx 5.5$ bpw stays within $\pm 0.4\%$ of BF16 WikiText-2 perplexity on every model, matching the SmoothQuant W8A8 quality envelope at $32\%$ fewer weight bits. Joint 2D coding outperforms polar (amplitude $\times$ phase) coding by 2--15pp $Δ$PPL at equal bitrate, and paired KL against BF16 tracks $Δ$PPL% at Spearman $ρ= 0.99$ across 37 (method, model) rows, consistent with a monotone composite bound from codec distortion to KL divergence. A 3.5bpw variant is competitive on quantization-tolerant architectures. At strict 4bpw, the rotated-codebook frontier method QTIP outperforms QAM-W; the contribution is the quality-preserving 5--6bpw band.

Narrowly finetuned language models memorize implanted content verbatim, but auditing what a deployed model has been taught, without access to its weights or training data, remains an open challenge. Recent work shows that activation differences between base and finetuned models carry readable traces of the finetuning domain; the state-of-the-art Activation Difference Lens (ADL) recovers a vague domain-level description but requires full "white-box" access to model internals. We introduce Contrastive Decoding Diffing (CDD), a model diffing method that operates on output-level logit distributions only, with no weight access, no layer selection, and no per-model tuning, yet recovers implanted facts. CDD consists of three ideas: bypassing the chat template to expose the raw finetuning prior, seeding generation with maximally vague pre-fills, and amplifying the logit-space difference between finetuned and base models at each decoding step. A single default configuration recovers implanted facts verbatim -- exact drug names, vote counts, physical measurements, and procedural details -- across four architectures (1B--32B parameters), uniformly outperforming ADL despite less access and running ~170x faster. Furthermore, CDD surfaces unintended data pipeline artifacts: a fictional persona introduced by the LLM data generator via mode collapse leaked into model weights and was extracted by CDD, constituting to our knowledge the first demonstrated end-to-end fingerprinting chain from data generator artifact to model weights to recovered output. We validate on real-domain finetuning settings, achieving near-perfect recovery across all single-dataset non-CoT variants and correctly identifying all four datasets in the mixed-dataset setting. CDD's success as a grey-box method outperforming white-box baselines underscores its practical utility for transparency and accountability in AI systems.

In the paper we show that there is an analogy between the operations occurring in a layer of a transformer (projections and layer normalizations, disregarding the feedforward neural network) and a step in the power method. Coherently with this analogy, we show that passing through a layer the tokens tend to be tilted towards the principal eigenvector of a matrix which is the product of the output and value weight matrices of that layer. In the special case of a transformer with shared weights (i.e., in which all layers have identical weights) then the alignment with this principal eigenvector is particularly evident empirically, and can also be shown analytically. The analogy also suggests a method to steer the output of the transformer towards an arbitrary desired direction in token space.

We apply the influence-adaptive Walsh geometry of a companion theory paper (arXiv:2605.01637) to extreme low-bit weight-only LLM quantization. The recipe is one math-invariant transformation: WHT-rotate each linear layer's weight matrix and rescale its columns by per-coordinate Walsh-basis activation energy before handing off to a reconstruction-error quantizer (Intel auto-round). This biases per-group integer rounding toward high-spectral-energy channels. On four pretrained decoder-only models from 135M to 1.5B parameters, BBT-spectral reduces wikitext-2 perplexity by 15-58% relative to vanilla auto-round at W2A16; we also report a TinyLlama-1.1B auxiliary data point. Three extensions transfer the recipe to families it failed on: a per-head PCA matrix-Gamma replacement of q_norm/k_norm for Qwen3 attention (PPL 136.76 -> 88.99 on Qwen3-0.6B); an SO(2) per-pair rotation that commutes with RoPE (PPL 36.93 -> 21.84 on Qwen2.5-1.5B); and an MoE-aware input-side absorption fix identified by architectural fuzzing of Laguna-style fused-expert layouts. A W2-vs-W4 ablation gives a deliberate negative control: the redistribution payoff falls within the +/-0.5 PPL noise floor at W4, consistent with the Schur-convexity intuition that the cost of unconcentrated influence vanishes as the noise budget shrinks. All quantized weights export to OpenVINO IR and run on Intel NPU + Arc dGPU + CPU with PPL invariant to device within +/-0.1. We do not claim a formal Boolean-to-real-valued transfer of the theory paper's majorization argument: the WHT activation energy used here is not the Boolean influence of the theory paper, the link is intuitive, and the contribution is engineering value rather than a transferred theorem. Head-to-head benchmarks against SpinQuant, QuaRot, QuIP-sharp, AQLM, OmniQuant, and ButterflyQuant at matched calibration are the main future-work item.

Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate $\frac{1}{2}$ for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, showing that the obtained convergence rates extend beyond the considered setting.

Neural network weights are increasingly a bottleneck for deployment, yet most compression pipelines treat layers independently and overlook cross-layer redundancy induced by function-preserving symmetries. We propose Motion-Compensated Weight Compression (MCWC), a weight-only codec that aligns permutation-symmetric blocks (e.g., hidden units and attention heads) to maximize cross-layer correspondence, turning depth into a predictable sequence. In the aligned coordinate system, MCWC uses a lightweight layer-sequential predictor with periodic keyframes and encodes only quantized prediction residuals using a learned entropy model trained under a rate distortion objective. A simple decoder reconstructs deployable weights by entropy decoding, dequantization, predictor-driven reconstruction, and inverse alignment, enabling fast weight materialization for inference. Across Transformer language modeling and vision classification, MCWC improves the rate accuracy Pareto frontier over strong quantization and learned weight-codec baselines, while maintaining competitive decode time. Ablations confirm that alignment, prediction, entropy modeling, and keyframe scheduling are each necessary for the full gains. Our code is available via https://github.com/Ism-ail11/MCWC.

Efficiently updating Large Language Models (LLMs) with new or evolving factual knowledge remains a central challenge, as even parameter-efficient adaptation can erode previously acquired reasoning abilities. This tension reflects a plasticity-stability dilemma: models must incorporate new knowledge while preserving skill-critical representations. In this work, we study this trade-off through the spectral structure of multilayer perceptron weight matrices. We show, both theoretically and empirically, that information essential for reasoning is not localized only in dominant singular directions, but is instead distributed across the singular spectrum. Motivated by this observation, we introduce PALoRA, a two-stage framework for knowledge injection with reduced interference. PALoRA first trains a Singular Value Fine-Tuning (SVF) expert on a reasoning dataset and uses its learned singular scaling vector as a frozen geometric probe to identify components that are critical for the target skill. It then performs factual knowledge injection with Low-Rank Adaptation (LoRA) under a structural orthogonality constraint, ensuring that updates avoid the identified skill-relevant subspace. Across Llama 3.1 8B and Mistral 7B, and across mathematical, coding, and scientific reasoning benchmarks, PALoRA preserves on average 95% of the SVF expert's reasoning performance while maintaining competitive factual recall. It consistently improves skill retention over prior spectral Parameter-Efficient Fine-Tuning (PEFT) methods while adding less than 0.006% parameter overhead.

Input-convex neural nets parametrize the convex potentials of density models and transport maps, and their convexity requires the inter-layer weights to be non-negative. Projected gradient descent enforces this by projecting after each step, and due to mini-batch noise the boundary is re-crossed indefinitely, which leads to an active set the projection never identifies. The differentiable alternative, direct softplus, optimizes a free latent weight through a softplus positivity map whose derivative attenuates the gradient exponentially where the weight is negative---the shoulder---so a coordinate that reaches it stays for an exponentially long time. To keep this unconstrained parametrization without its slow escape, we propose the lift, which replaces the free latent weight by a learnable slack plus an unconstrained network---the body---that takes a permutation-invariant summary of the training batch as input. The latent weight thus varies with the batch before the positivity map, and couples to the gradient formed on it. We show that this coupling enters the variance of the update to the latent weight at first order in the fluctuation, and that the slack, the batch dependence and the shared batch are each needed for it to act. Where the coupling aligns positively with the loss curvature, that variance is larger under the lift than under direct softplus, and a coordinate leaves the shoulder sooner. We compare the lift with the two existing methods on several applications. Where a constrained weight of direct softplus reaches the shoulder and does not leave, the lift fits the target more closely and reaches the same reconstruction about three times sooner. Where almost none reaches it, the methods agree.

Large language models (LLMs) offer a scalable mechanism to elicit domain-informed prior information for high-dimensional variable selection. However, existing methods such as LLM-Lasso are sensitive to weight quality, with performance degrading substantially when LLM-generated weights are inaccurate. To address this challenge, we first introduce a framework for quantifying the quality of LLM-generated weights, enabling rigorous evaluation of LLM-informed methods across varying weight regimes. We then propose the LLM Sparsity Prior (LSP), which integrates LLM-generated weights into the prior inclusion probabilities of Spike-and-Slab and Spike-and-Slab Lasso models via two interpretable hyperparameters governing global sparsity and weight concentration. Hierarchical hyperpriors on these parameters allow the model to dynamically discount uninformative or misleading weights, improving robustness without sacrificing gains when weights are accurate. Finally, we develop principled prompt engineering strategies and validate the method on a private medical dataset studying Acute Kidney Injury. LSP improves prediction accuracy and identifies clinically relevant features missed by the baselines, with robustness to prompt variation and particular effectiveness in low-data regimes.

Agent orchestration frameworks have proliferated, collectively exceeding 290,000 GitHub stars across LangGraph, CrewAI, Google ADK, OpenAI Agents SDK, Semantic Kernel, Strands, and LlamaIndex. All follow the same pattern: an external orchestrator above the LLM, injecting instructions and routing decisions every turn. Recent work has shown this architecture is dominated for procedural tasks by simply providing the procedure in a frontier model's system prompt [Dennis et al., 2026a], at the cost of consuming the context window, requiring a frontier model for every conversation, and exposing proprietary procedures to third-party providers. Compiling the procedure into the weights of a small fine-tuned model -- creating a subterranean agent -- should resolve all of these concerns, and prior work (SimpleTOD, FireAct, SynTOD, WorkflowLLM, Agent Lumos) has shown the technique works. Yet developer adoption has overwhelmingly favored orchestration. We identify three perceived barriers and address each empirically across travel booking (14 nodes), Zoom support (14 nodes, product-specific knowledge), and insurance claims (55 nodes, 6 decision hubs).

Scaling laws have made language-model performance predictable from model size, data, and compute, but they typically treat the optimizer as a fixed training detail. We show that this assumption misses a fundamental axis of representation scaling: how effectively the optimizer converts added FFN width into utilized spectral capacity. Using eigenspectra of feed-forward network representations, measured through soft and hard spectral-ranks, we find that \emph{the same Transformer architecture realizes markedly different spectral scaling laws when trained with different optimizers}. Holding architecture and width schedule fixed, AdamW exhibits weak hard-rank scaling ($β$=0.44) on rare-token (TAIL) representations where learning is known to be hardest, whereas Muon achieves linear scaling ($β$=1.02) in the same regimes, a $2.3\times$ increase in the scaling exponent. This difference is not reducible to validation loss: AdamW configurations can match low-rank Dion variants in perplexity, under extended training, while exhibiting sharply different spectral geometry, demonstrating that matched loss does not imply matched representation structure. Hard--soft rank asymmetry further reveals that optimizers differ not only in how much capacity is realized, but also in how that capacity is structured across eigenmodes. To disentangle optimizer effects from architectural ones, we compare against architectural interventions (e.g., attention rank and positional encoding), and find that optimizer-induced spectral shifts often exceed the architectural effects. These results suggest optimization as a first-class axis of representation scaling, motivating optimizer--architecture co-design.

Deep neural networks have achieved impressive performance on a variety of tasks, but their brittleness to distributional shifts remains a significant barrier to real-world deployment. In this paper, we propose a framework to analyse and quantify the distributional robustness of neural networks by studying the interactions between layer weights and activations. We model these interactions using Bernoulli distributions, using the separation between classes as a diagnostic proxy for robustness. We demonstrate the usefulness of this framework through models trained on CIFAR-10 and ImageNet. We show that our proposed metrics can distinguish between networks that have memorised their training data and those that have not. We also perform analogous experiments in the activation space and find that the same properties do not hold up. Additionally, we investigate the behaviour of our metrics under various distribution shifts and show that these shifts reduce separation under our path-based diagnostics. Our results suggest that this framework provides useful model-level diagnostics of representation structure and robustness.

Transformers trained on modular arithmetic exhibit sharp transitions between memorization, generalization, and collapse. We show that weight decay acts as a scalar empirical control parameter for these regimes, and introduce two cheap online diagnostics, mean pairwise attention-head cosine similarity and entropy standard deviation, that track training dynamics from attention activations alone and complement loss-landscape diagnostics at lower compute cost. Across eleven experimental conditions and three model scales (0.82M to 85M parameters), the weight-decay axis separates memorization, developmental grokking, and collapse. A near-transition logistic fit localizes the memorization-to-developmental boundary at $λ_c=0.0158$ (95% CI [0.0109, 0.0200], N=210); a power-law fit gives an empirical exponent $ν=0.757$ (CI [0.725, 0.799]). Reference exponents $ν=1/2$ and 3D Ising $ν\approx 0.63$ lie outside this empirical CI under our four-bin grid, so we report $ν$ as empirical and defer universality-class identification to denser finite-size-scaling work. A horizon-matched multi-task replication (n=280, four modular operations) preserves the weight-decay control pattern; a paired attention-head re-initialization experiment at $λ=0.05$ changes Phase-2 amplitude (Cohen's $d=-1.190$, n=10, $p_t=4.5 \times 10^{-3}$), while matched weight-norm clipping does not. Three cross-architecture probes (4L MLP, 4L LSTM, and 4L Mamba; each n=70) replicate the weight-decay-controlled transition with architecture-specific $λ_c$ values. Main diagnostic claims are scoped to modular arithmetic in small transformer attention models; the non-attention experiments are scope probes, and architecture-wide, language-model, and universality-class claims are out of scope.

Deep learning methods have demonstrated promising results in predicting BI-RADS scores from mammography images. However, the interpretation of these images can vary, leading to discrepancies even among radiologists. Given the inherent complexity of mammograms, training classification models solely on image labels often yields limited performance. To address this challenge, we curated 2313 mammogram images and their corresponding captions from two mammography atlases. Our proposed approach employs a multi-modal model that uses a pretrained PubMedBERT as the language component. By training this model on image-text pairs with contrastive learning, we enable the vision encoder to absorb the rich information contained in the captions, thereby improving its understanding of mammography findings. We then fine-tune the vision encoder on two datasets for BI-RADS prediction, achieving superior performance compared with models trained without this pretraining, particularly when labeled samples are scarce. The improvement in the 3-class average F1 score ranges from +1% to +14%: a +1% increase with 40K training samples, and a +14% increase with 1K samples. Furthermore, our experiments reveal that 2K image-text pairs from mammography atlases can be more informative than 2K labeled samples for label prediction, with an average margin of +1.1% when more than 10K training samples are available. Overall, our work provides a vision-language model for mammography and highlights the value of textual information from mammography atlases. In addition, we publicly release preprocessed mammography images of the TEKNOFEST dataset. The training code, pre-trained model weights, data extraction scripts, and the released dataset are publicly available at: https://github.com/igulluk/MAM-CLIP

Neural network checkpoints have quietly become a large-scale data resource: millions of trained weight vectors now exist, each encoding task-, domain-, and architecture-specific knowledge. This position paper argues that model checkpoints should be treated as a first-class data modality, and that generative modeling in weight space should be standardized as a core machine learning primitive. Recent advances demonstrate that neural weights can be synthesized on demand, often matching fine-tuning performance while reducing adaptation cost by orders of magnitude. We contend that these results reflect an underlying structural fact: high-performing models occupy low-dimensional, highly structured regions of weight space shaped by symmetry, flatness, modularity, and shared subspaces. Building on this view, we organize existing methods into a five-stage pipeline, survey applications where the approach is already practical, and clarify current limits: adapter-scale and conditional generation are advancing rapidly, while unrestricted frontier-scale checkpoint synthesis remains open. Our goal is to shift the community's default mindset from optimizing models per task to sampling models from learned weight distributions, accelerating toward an era in which AI systems routinely improve or create other AI systems.

Pre-norm Transformers with RMSNorm tolerate ternary {-1,0,+1} weight quantization with surprisingly small loss (Ma et al., 2024). We give a geometric explanation via sign-magnitude decomposition of weight perturbations. In a two-layer ReLU + RMSNorm model with i.i.d. Gaussian weights, sign-flips produce $π/(π-2) \approx 2.75$ times more transverse output energy than sign-preserving magnitude perturbations of equal Frobenius norm, as the flip rate $p \to 0$ (Theorem 3). The mechanism: ReLU creates a hidden-space directional asymmetry between the two perturbation types, which RMSNorm's transverse-projection Fréchet derivative selectively exposes. Sign-quantization error is itself a sign-preserving perturbation with angular alignment $\cos^2 \to 2/π$ (Theorem 4); its post-ReLU radial fraction ($0.365$) matches the pre-ReLU value $1-2/π$ within $0.4\%$, so ReLU is approximately transparent to ternary error. Multi-layer compounding of the $2.75\times$ factor is not experimentally supported; the gap to real-model sign sensitivity arises from outlier features violating delocalization. For an input dimension with amplitude $α$, a single sign-flip produces post-ReLU energy amplified by $R \approx nα^2$ relative to a delocalized entry. On TinyLlama-1.1B, at linear response ($p \leq 0.5\%$), count-matched NLL leverage stabilizes at $\sim 10\times \approx n\mathbb{E}[α^2]$, matching the per-entry theory; the all-column NLL ratio of $5.0\times$ falls within $R_{\mathrm{col}} \leq 19$ ($67\times$ PPL gap reflects metric nonlinearity). Measured outlier $α$ at layer 12 (median $0.024$, max $0.26$) confirms heavy-tailed concentration. The Bussgang constant $2/π$, RMSNorm geometry, and ReLU half-space structure together explain sign-magnitude asymmetry in pre-norm models, with $R \propto nα^2$ accounting for real-model deviations.

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible update rules consistently improve final validation loss, reduce expert load imbalance in sparse MoE models, and in several cases control final vocabulary-logit growth, improve router stability, and overall training stability over the corresponding AdamW updates.

Muon has recently emerged as a competitive alternative to AdamW for large-scale pre-training, with orthogonalization via Newton-Schulz (NS) iterations as its core operation. Existing Muon variants apply a uniform NS schedule to all parameter matrices, overlooking possible differences in orthogonalization difficulty and its impact on performance. Through a systematic empirical study, we show that this per-matrix heterogeneity is pervasive and largely determined by matrix geometry, which evolves dynamically across operator types, training stages, and network depths. As a result, uniform NS schedules can lead to uneven orthogonalization quality across the model. Motivated by these findings, we propose Adaptive Muon Orthogonalization (AMO), an observe-then-commit method that measures weight geometry by operator type early in training and then uses these signals to allocate the NS budget for the remainder of training. AMO delivers consistent improvements over uniform-schedule Muon across standard, prolonged, and continual pre-training, surpassing the strongest baseline by +0.76 on Llama3.1-1.4B and +0.51 on Qwen3-1.7B in average downstream performance of 12 evaluation tasks.

The design of modern neural architectures has converged through incremental empirical choices, yet the mechanisms governing their training dynamics remain only partially understood. We identify and analyze a negative weight drift induced by the interaction between standard losses and positively biased activation functions. We prove that under MSE or cross-entropy loss, the gradient with respect to positive pre-activations is non-negative in expectation at initialization, driving downstream weights toward negative values during early training. The drift is intrinsic to optimization rather than data, and persists across architectures (MLP, ResNet, ViT, GPT-nano, MP-SENe) and asymmetric activation functions (ReLU, GELU, SiLU). Coupled with ReLU, weight drift produces activation sparsity reaching up to 90% in GPT-nano. We characterize the sparsity-accuracy tradeoff across 79 configurations and identify a sharp accuracy cliff above $\sim$70% activation sparsity. While ReLU$^2$ achieves a good sparsity--accuracy ratio in GPT-nano, it pathologically amplifies identified activation spikes in intermediate transformer layers. Clipping resolves this while preserving the representational benefits of squaring: clipped ReLU$^2$ outperforms its unclipped version, and GELU$^2$ achieves the lowest validation loss on GPT-nano. Code is available at https://github.com/On-Point-RND/BugOrFeature.

We apply the Weibull distribution -- a two-parameter family from extreme-value theory -- as a diagnostic framework for element-wise weight magnitude distributions in transformers. At initialization, i.i.d. Gaussian weights give |w| ~ HalfNormal, yielding k ~ 1.20 via middle-80% probability-plot fit (the protocol used throughout this work). This anchor makes k a principled, architecture-independent measuring stick for training dynamics; fitting each weight matrix independently at every layer at every checkpoint enables per-component, per-layer, and per-step diagnostics that aggregate statistics cannot resolve. Applying this framework to 12 model entries spanning 7 architectural families (Pythia, OLMo-1/2, LLaMA-3, Mistral, Qwen2.5/3) reveals three findings. First, FFN modules and the attention output projection W_o -- the Transmission Class -- fall in a narrow k band: median terminal k in [1.186, 1.204] across 12 entries (cross-family CV = 0.51%), shared across SwiGLU/GeLU activations, Pre-LN/QK-Norm placements, and 70M-14B sizes. Second, the attention input projections W_q, W_k -- the Selection Class -- depart from the Weibull family, with severity shaped by storage: separately-stored Q/K (OLMo-1, OLMo-2) yields k in [0.76, 0.99] (deep); GQA models yield k in [1.10, 1.16] (mild); Pythia's merged W_qkv occupies a transitional zone tracking training budget T/tau monotonically. Third, lambda grows substantially during training and scales with sqrt(eta/lambda_wd) within the Pythia family (Pearson r = 0.94, three Transmission kinds), directionally consistent with Fan et al. (2025). The two parameters carry independent information: k labels the functional class, lambda labels training progress. We release npm-weibull-py v0.4 (Python library) and DATABASE_v9_1 at https://github.com/tiexinding/NPM-Weibull-public .

We introduce an empowered transposed Fully Connected Weighted (t-FCW) graph representation to embed point clouds into a metric space. While original t-FCW has shown promising results for point cloud classification, the reasons behind its effectiveness and its broader applicability remained unclear. In this work, we analyze the properties that make the empowered and original t-FCW effective and design a network that uses the empowered t-FCW exclusively as feature extractors. From an interpretability perspective, we build memory banks for classification, part segmentation, and semantic segmentation using the empowered t-FCW. Our analysis reveals that the empowered t-FCW inherits robustness from surface descriptors, provides interpretability through dimension-wise relations. These properties enable a highly efficient and interpretable network, which processes the ModelNet40 classification problem in approximately 7 seconds on an NVIDIA RTX A5000 GPU. Importantly, empowered t-FCW can function both as a lightweight standalone baseline and as a complementary plug-in to existing deep models.

Hyperparameter transfer across model architectures dramatically reduces the amount of compute necessary for tuning large language models (LLMs). The maximal update parameterization (μP) ensures transfer through principled mathematical analysis but can be challenging to derive for new model architectures. Building on the spectral feature-learning view of Yang et al. (2023a), we make two advances. First, we promote spectral norm conditions on the weights from a heuristic to the definition of feature learning, and as a consequence arrive at the Complete-P depth and weight-decay scalings without recourse to lazy-learning. Second, we consider a modified spectral norm that preserves the valid scaling law of network weights when weight matrices are not full rank. This enables (to our knowledge, the first) derivation of μP scalings for grouped-query attention (GQA). We demonstrate the efficacy of our theoretical derivations by showing learning rate transfer across the GQA repetition hyperparameter as well as experiments regarding transfer over weight decay.

Scaled Outer Product (SOP) is a post-training quantization methodology for large language model weights, designed to deliver near-lossless fidelity at 4.5--6 bits per weight on hardware with per-layer LUT decode. The methodology combines per-layer search of fixed and dynamic codebook pairs selected by a per-block selection bit, signed per-block scales, activation-weighted cosine selection, and multiple-choice knapsack promotion of sensitive layers with outlier and sparse-residual correction. Fixed codebooks include NF4, BOF4, Split87, and SH4; per-layer optimized codebooks (DD4) are hosted in LUT SRAM. A new hardware-efficient LUT output format (HIF) is proposed to improve performance, energy, and cost. Across six open model families, the recommended FP6 operating point (E2M3sUE4M4, 6.5 bpw) achieves lower weight reconstruction error than the conventional per-layer-POT FP8 baseline (E4M3, 8.0 bpw) at 1.5 bpw lower storage cost, demonstrating that block-scaled small atoms with carefully chosen scale precision can replace conventionally-deployed FP8. Full evaluation across the 4.5--6 bpw range, including layer promotion and sparse residual correction, is reported in a companion paper.

Large language models (LLMs) are pretrained by minimizing the cross-entropy loss for next-token prediction. In this paper, we study whether this optimization strategy can induce geometric structure in the learned model weights and context embeddings. We approach this problem by analyzing a constrained layer-peeled optimization program, which serves as a mathematically tractable surrogate for LLMs by treating the output projection matrix and last-layer context embeddings as optimization variables. Our analysis of this nonconvex optimization program demonstrates that symmetries in the target next-token distributions are transferred to the global minimizers of the layer-peeled model in a precise group-theoretic sense. Specifically, we prove that when the target tokens exhibit a cyclic-shift symmetry (such as the seven days of the week or the twelve months of the year), the optimal logit matrix is exactly circulant, and the Gram matrices of both the output projections and the context embeddings form circulant geometries as well. Next, for exchangeable target distributions invariant under the symmetric group and, more generally, under two-transitive group actions, we show that the global optimal output projection matrix forms a simplex equiangular tight frame, while the optimal logit matrix and context embeddings inherit the permutation symmetries present in the input data. A key technical step is to reduce the constrained nonconvex factorized problem to an explicit logit-level convex characterization for cyclic symmetry and to a symmetry-based lower bound for permutation symmetry, together with a sharp characterization of the optimal factorization. Finally, we empirically demonstrate that open-source LLMs naturally exhibit symmetries consistent with our theoretical predictions, despite being trained without any explicit regularization promoting such geometric structure.

Randomized neural networks (RdNNs) enable efficient, backpropagation-free training by freezing randomly initialized input-to-hidden weights, which permits a closed-form solution for the output layer. However, conventional random initialization is blind to inter-feature dependence, ignoring correlations, asymmetries, and tail dependence in the data, which degrades conditioning and predictive performance. To the best of our knowledge, this limitation remains unaddressed in the RdNN literature. To close this gap, we propose CAWI (Copula-Aligned Weight Initialization), a framework that draws input-to-hidden weights from a data-fitted copula that matches empirical dependence, ensuring the frozen projections respect inter-feature dependence without sacrificing the closed-form solution. CAWI (i) maps each feature to the unit interval using empirical CDFs, (ii) fits a multivariate copula that captures rank-based dependence among features, and (iii) samples each weight column w_j from the fitted copula and applies a fixed inverse marginal transform to set scale. The objective, solver, and "freeze-once" paradigm remain unchanged; only the sampling law for W becomes dependence-aware. For dependence modeling, we consider two copula families: elliptical (Gaussian, t) and Archimedean (Clayton, Frank, Gumbel). This enables CAWI to handle diverse dependence, including tail dependence. We evaluate CAWI across 83 diverse classification benchmarks (binary and multiclass) and two biomedical datasets, BreaKHis and the Schizophrenia dataset, using standard shallow and deep RdNN architectures. CAWI consistently delivers significant improvements in predictive performance over conventional random initialization. Code is available at: https://github.com/mtanveer1/CAWI

Quantization is an effective strategy to reduce the storage and computation footprint of large language models (LLMs). Post-training quantization (PTQ) is a leading approach for compressing LLMs. Popular weight quantization procedures, including GPTQ and RTN, suffer in model utility, especially at aggressive quantization levels (sub-4-bit). We propose ADMM-Q, a novel weight quantization algorithm that considers the layer-wise quantization problem. Our algorithm is based on a combinatorial variant of the Alternating Direction Method of Multipliers (ADMM). Our operator-splitting procedure updates weights continuously to minimize the layer-wise reconstruction error, while gradually enforcing the quantization constraints with convergence guarantees. We propose additional algorithmic enhancements (e.g., penalty scheduling, preconditioning, and a local search post-processing step) to make ADMM-Q efficient at LLM scale. ADMM-Q is modular and can be used as a drop-in replacement for any weight quantizer within existing quantization pipelines: ADMM-Q is fully composable with existing techniques including range clipping, learned or random rotations, and activation scaling. Using ADMM-Q in place of GPTQ on Qwen3-8B, we decrease WikiText-2 perplexity in: (i) the W3A16 weight-only setting (12.85 $\rightarrow$ 10.06); (ii) the W4A8 SmoothQuant procedure (9.29 $\rightarrow$ 8.68); and (iii) the W2A4KV4 SpinQuant procedure (66.11 $\rightarrow$ 19.42).

Why does weight decay work? We prove that, in any fixed-precision regime, the smallest weight norm of a looped neural network outputting a binary string equals the Kolmogorov complexity of that string, up to a logarithmic factor. This implies that weight decay induces a prior matching Solomonoff's universal prior, the optimal prior over computable functions, up to a polynomial factor. The result is norm-agnostic: in fixed precision, every weight norm collapses to the non-zero parameter count up to constants, so the same sandwich bound holds for any norm used as a regulariser. The proof has two short reductions: any program for a universal Turing machine can be encoded into neural weights at unit cost per program bit, and any fixed-precision network can be described by enumerating its non-zero parameters with logarithmic addressing overhead. Both bounds are tight up to constants, with the logarithmic factor realised by permutation encodings: a network whose parameters encode a permutation produces a string whose Kolmogorov complexity is the non-zero parameter count times its logarithm. The fixed-precision assumption is essential: with infinite precision, neural networks can encode non-computable functions and the weight norm loses its relevance.

Muon has emerged as a strong competitor to AdamW for language model pre-training, yet its behavior at scale is sensitive to weight decay. Recent work has observed that, for Muon without decoupled weight decay, the spectral norm of weight matrices drifts upward over training. Through a decomposition of the spectral norm into a row-magnitude factor and a row-coherence factor, we identify the former as the empirical driver of this drift under Muon, while the latter remains well-behaved along the trajectory. Motivated by this diagnosis, we introduce Muown, a drop-in replacement for Muon that treats the row-magnitude vector as an explicit optimizer variable, updating it under the $\ell_\infty$ geometry induced by the decomposition, while applying Muon unchanged to the remaining direction component. We prove that Muown attains the optimal non-convex rates in both deterministic and stochastic regimes under a dual norm aligned with the underlying geometries and with a stochastic noise coefficient that empirically remains below that of Muon throughout training. Across GPT-style pre-training on FineWeb-Edu with model sizes from 124M up to 2.7B parameters, Muown improves perplexity over Muon, SOAP, AdamW, and Lion. It also widens the plateau of near-optimal learning rates across model scales, reduces sensitivity to weight decay, and avoids the spectral norm drift at negligible step-time overhead when appropriately sharded.

The quality of open-weight language models has dramatically improved in recent years. Sharing weights greatly facilitates model adoption by enabling their use across diverse hardware and software platforms. They also allow for more open research and testing, to the extent that users can use them as checkpoints, fine-tune them according to their needs, and potentially redistribute them. In some cases, however, concerns on modifying these weights towards unauthorized uses may outweigh the pros of giving users such a freedom. Defending against such adaptation is non-trivial: since an adaptive attacker can observe all weights and architectures by definition, they can reverse simple structural defenses, and use optimization to defeat the simplest locking mechanisms. In this work, we exploit the inference-training asymmetry of automatic differentiation as a novel defense axis. We propose DLR-Lock, a method where the purveyor of the model purposely replaces each pretrained MLP in their model with a deep low-rank residual network (DLR-Net) of comparable parameter count, forcing activation memory that grows linearly with depth during backpropagation. DLR-Nets are efficiently trained via module-wise distillation. We show that, beyond this memory overhead, DLR-Lock results in architectural mismatches that complicate the optimization landscape of standard fine-tuning, and a backward pass that incurs disproportionately more overhead than the forward pass. Our defense succeeds in withstanding adaptive attackers with full knowledge of the defense strategy while preserving the original model's capabilities. Experiments on LLM validate these claims.

A surprising phenomenon in the training of neural networks is the ability of gradient descent to find global minimizers of the training loss despite its non-convexity. Following earlier works, we investigate this behavior for wide shallow networks. Existing results essentially cover the case of ReLU activations and the case of sigmoid activations with scalar output weights. We study a large class of models that includes multi-head attention layers and two-layer sigmoid networks with vector output weights. Building upon [Chizat and Bach, 2018], we prove that all non-global minimizers of the training loss are unstable under gradient descent dynamics. Thus, when the initial distribution of the parameters has full support (which includes the popular Gaussian case), and in the many hidden neurons or attention heads limit, continuous-time gradient descent can only converge to global minimizers. Establishing the instability of non-global minimizers corresponds to the construction of an ``escaping active set'' -- we complete the proof of [Chizat and Bach, 2018] to construct this set for models with bounded nonlinearities and scalar output weights. We also extend this construction to new cases for models with vector output weights. Finally, we show the well-posedness and the stability with respect to discretization of the mean field training dynamic for sub-Gaussian initializations.

Macroscopic transport modelling aims to predict traffic flows after proposed public policy interventions, such as a new road or railway section or a temporary road closure. As such, it is a vital step in infrastructure planning and development. Traditionally, building a transport model has relied on complex understanding of socio-economic characteristics of the population requiring expensive data collection via surveys, which are prone to biases. Previous numerical frameworks to optimize transport models to fit observed traffic flows are not easily-interpretable and can lead to overfit. We present here an alternative: a data-driven modelling protocol with objective function based on traffic counts, which can be nowadays cheaply and reliably obtained; explainable model weights; and a controlled path to increase model complexity and accuracy. We demonstrate our approach on several toy and realistic examples, and suggest ways to generalize to multimodal systems including public transport.