Daily Papers

We show that robustness to post-training quantization (PTQ) is a transferable direction in weight space. We call this direction the quantization vector: extracted from a donor task by simple weight-space arithmetic, it can be used to patch a receiver model and improve post-PTQ Top-1 accuracy by up to 60 points in a 3-bit setting, without receiver-side quantization-aware training (QAT). Because the method requires no receiver training data, it provides a zero-shot, low-cost alternative to QAT for extremely low-bit deployment. Across four ViT scales and 22 image classification tasks, donor quantization vectors often yield substantial gains even when donor and receiver tasks differ markedly. We further prove rigorously that quantization vectors are well-defined and do not suffer from reparameterization symmetries, and provide a local geometric account of their effect. Together, these results suggest that quantization robustness can be partially isolated, reused, and transferred through simple weight-space algebra.

Providing high-quality feedback to Large Language Models (LLMs) on a diverse training distribution can be difficult and expensive, and providing feedback only on a narrow distribution can result in unintended generalizations. To better leverage narrow training data, we propose contrastive weight steering, a simple post-training method that edits the model parameters using weight arithmetic. We isolate a behavior direction in weight-space by subtracting the weight deltas from two small fine-tunes -- one that induces the desired behavior and another that induces its opposite -- and then add or remove this direction to modify the model's weights. We apply this technique to mitigate sycophancy and induce misalignment, and find that weight steering often generalizes further than activation steering, achieving stronger out-of-distribution behavioral control before degrading general capabilities. We also show that, in the context of task-specific fine-tuning, weight steering can partially mitigate undesired behavioral drift: it can reduce sycophancy and under-refusals introduced during fine-tuning while preserving task performance gains. Finally, we provide preliminary evidence that emergent misalignment can be detected by measuring the similarity between fine-tuning updates and an "evil" weight direction, suggesting that it may be possible to monitor the evolution of weights during training and detect rare misaligned behaviors that never manifest during training or evaluations.

Task arithmetic has recently emerged as a cost-effective and scalable approach to edit pre-trained models directly in weight space: By adding the fine-tuned weights of different tasks, the model's performance can be improved on these tasks, while negating them leads to task forgetting. Yet, our understanding of the effectiveness of task arithmetic and its underlying principles remains limited. We present a comprehensive study of task arithmetic in vision-language models and show that weight disentanglement is the crucial factor that makes it effective. This property arises during pre-training and manifests when distinct directions in weight space govern separate, localized regions in function space associated with the tasks. Notably, we show that fine-tuning models in their tangent space by linearizing them amplifies weight disentanglement. This leads to substantial performance improvements across multiple task arithmetic benchmarks and diverse models. Building on these findings, we provide theoretical and empirical analyses of the neural tangent kernel (NTK) of these models and establish a compelling link between task arithmetic and the spatial localization of the NTK eigenfunctions. Overall, our work uncovers novel insights into the fundamental mechanisms of task arithmetic and offers a more reliable and effective approach to edit pre-trained models through the NTK linearization.

As an efficient alternative to conventional full finetuning, parameter-efficient finetuning (PEFT) is becoming the prevailing method to adapt pretrained language models. In PEFT, a lightweight module is learned on each dataset while the underlying pretrained language model remains unchanged, resulting in multiple compact modules representing diverse skills when applied to various domains and tasks. In this paper, we propose to compose these parameter-efficient modules through linear arithmetic operations in the weight space, thereby integrating different module capabilities. Specifically, we first define addition and negation operators for the module, and then further compose these two basic operators to perform flexible arithmetic. Our approach requires no additional training and enables highly flexible module composition. We apply different arithmetic operations to compose the parameter-efficient modules for (1) distribution generalization, (2) multi-tasking, (3) unlearning, and (4) domain transfer. Additionally, we extend our approach to detoxify Alpaca-LoRA, the latest instruction-tuned large language model based on LLaMA. Empirical results demonstrate that our approach produces new and effective parameter-efficient modules that significantly outperform existing ones across all settings.

Changing how pre-trained models behave -- e.g., improving their performance on a downstream task or mitigating biases learned during pre-training -- is a common practice when developing machine learning systems. In this work, we propose a new paradigm for steering the behavior of neural networks, centered around task vectors. A task vector specifies a direction in the weight space of a pre-trained model, such that movement in that direction improves performance on the task. We build task vectors by subtracting the weights of a pre-trained model from the weights of the same model after fine-tuning on a task. We show that these task vectors can be modified and combined together through arithmetic operations such as negation and addition, and the behavior of the resulting model is steered accordingly. Negating a task vector decreases performance on the target task, with little change in model behavior on control tasks. Moreover, adding task vectors together can improve performance on multiple tasks at once. Finally, when tasks are linked by an analogy relationship of the form ``A is to B as C is to D", combining task vectors from three of the tasks can improve performance on the fourth, even when no data from the fourth task is used for training. Overall, our experiments with several models, modalities and tasks show that task arithmetic is a simple, efficient and effective way of editing models.

Machine unlearning aims to selectively remove specific knowledge from a trained model. Existing approaches, such as Task Arithmetic, fine-tune the model on the forget set to create a task vector (i.e., a direction in weight space) for subtraction from the original model's weight. However, their effectiveness is highly sensitive to hyperparameter selection, requiring extensive validation to identify the optimal vector from many fine-tuned candidates. In this paper, we propose a novel method that utilizes all fine-tuned models trained with varying hyperparameters instead of a single selection. Specifically, we aggregate the computed task vectors by retaining only the elements with consistent shared signs. The merged task vector is then negated to induce unlearning on the original model. Evaluations on zero-shot and standard image recognition tasks across twelve datasets and four backbone architectures show that our approach outperforms state-of-the-art methods while requiring similar or fewer computational resources. Code is available at https://github.com/naver-ai/negmerge.

Task Arithmetic is a model merging technique that enables the combination of multiple models' capabilities into a single model through simple arithmetic in the weight space, without the need for additional fine-tuning or access to the original training data. However, the factors that determine the success of Task Arithmetic remain unclear. In this paper, we examine Task Arithmetic for multi-task learning by framing it as a one-shot Federated Learning problem. We demonstrate that Task Arithmetic is mathematically equivalent to the commonly used algorithm in Federated Learning, called Federated Averaging (FedAvg). By leveraging well-established theoretical results from FedAvg, we identify two key factors that impact the performance of Task Arithmetic: data heterogeneity and training heterogeneity. To mitigate these challenges, we adapt several algorithms from Federated Learning to improve the effectiveness of Task Arithmetic. Our experiments demonstrate that applying these algorithms can often significantly boost performance of the merged model compared to the original Task Arithmetic approach. This work bridges Task Arithmetic and Federated Learning, offering new theoretical perspectives on Task Arithmetic and improved practical methodologies for model merging.

Agent systems increasingly use textual skills to encode reusable task procedures, but injecting these skills into the prompt at every step incurs substantial context overhead and exposes skill content as plaintext. We present LatentSkill, a framework that converts textual skills into plug-and-play LoRA adapters through a pretrained hypernetwork. LatentSkill stores skill knowledge in weight space rather than context space, removing per-step skill tokens while preserving modular loading, scaling, and composition. On ALFWorld and Search-QA, LatentSkill outperforms the corresponding in-context skill baseline while using substantially fewer prefill tokens: it improves ALFWorld success by 21.4 and 13.4 points on the seen and unseen splits with 64.1% fewer prefill tokens, and improves Search-QA exact match by 3.0 points with 72.2% lower skill-token overhead. Further analysis shows that generated skill LoRAs form a structured semantic geometry, can be precisely controlled via the LoRA scaling coefficient, and can be composed through parameter-space arithmetic when skill components are aligned. These findings suggest that weight-space skills provide an efficient, modular, and less exposed substrate for extending LLM agents.

High-reliability long-horizon robotic manipulation has traditionally relied on large-scale data and compute to understand complex real-world dynamics. However, we identify that the primary bottleneck to real-world robustness is not resource scale alone, but the distributional shift among the human demonstration distribution, the inductive bias learned by the policy, and the test-time execution distribution -- a systematic inconsistency that causes compounding errors in multi-stage tasks. To mitigate these inconsistencies, we propose χ_{0}, a resource-efficient framework with effective modules designated to achieve production-level robustness in robotic manipulation. Our approach builds off three technical pillars: (i) Model Arithmetic, a weight-space merging strategy that efficiently soaks up diverse distributions of different demonstrations, varying from object appearance to state variations; (ii) Stage Advantage, a stage-aware advantage estimator that provides stable, dense progress signals, overcoming the numerical instability of prior non-stage approaches; and (iii) Train-Deploy Alignment, which bridges the distribution gap via spatio-temporal augmentation, heuristic DAgger corrections, and temporal chunk-wise smoothing. χ_{0} enables two sets of dual-arm robots to collaboratively orchestrate long-horizon garment manipulation, spanning tasks from flattening, folding, to hanging different clothes. Our method exhibits high-reliability autonomy; we are able to run the system from arbitrary initial state for consecutive 24 hours non-stop. Experiments validate that χ_{0} surpasses the state-of-the-art π_{0.5} in success rate by nearly 250%, with only 20-hour data and 8 A100 GPUs. Code, data and models will be released to facilitate the community.

AI Engram (Kwon et al., 2026) formalizes the four engram criteria of neuroscience as a constrained inverse problem in weight space and solves it closed-form: concept-specific memory traces become linear objects that can be extracted once and combined arithmetically. Appendix F states the Compositional Memory States Hypothesis: edited models live on "a commutative manifold where the integration of A and B reaches a consistent equilibrium regardless of the learning sequence." The evidence base is single and paired edits -- in materials terms, single-cycle tests, in which fatigue accumulation is structurally invisible. Whether the hypothesis holds under sequential load is exactly the "temporal dynamics" question the paper defers to future work. We run that test on the authors' own reference implementation, at their reported best edit strength (TOFU alpha=0.6, a choice favoring the linearity hypothesis), with pre-registered predictions, across three model charges (two vendors, two architecture families). Four findings replicate across all three: (1) zero-shot composition and sequential re-calibrated editing diverge by 61-71% of the edit magnitude; (2) cut order is not interchangeable, and the effect scales with concept overlap -- in one charge the order of cutting two Paris landmarks decides whether an uninvolved third concept survives; (3) the survivors' layer-input covariances -- the method's own sufficient statistics, read as strain gauges -- drift monotonically with every further cut, in every surviving concept, in every charge; (4) erased knowledge partially returns under subsequent unrelated cuts. Appendix F's commutative-manifold hypothesis is thereby falsified for sequential editing; the single-edit results of the original paper are untouched. For unlearning-as-compliance: erasure certified today does not certify the artifact after its next edit.

Model merging combines the parameters of multiple neural networks into a single model without additional training. As fine-tuned large language models (LLMs) proliferate, merging offers a computationally efficient alternative to ensembles and full retraining, enabling practitioners to compose specialized capabilities at minimal cost. This survey examines model merging in the LLM era through the FUSE taxonomy, organized along Foundations, Unification Strategies, Scenarios, and Ecosystem. We first establish the theoretical underpinnings of merging, including loss landscape geometry and mode connectivity, then systematically review the algorithmic space spanning weight averaging, task vector arithmetic, sparsification-enhanced methods, mixture-of-experts architectures, and evolutionary optimization. We further examine downstream applications across multi-task learning, safety alignment, domain specialization, and federated learning, and survey the supporting ecosystem of tools and evaluation benchmarks. Finally, we identify key open challenges and future directions, aiming to equip researchers and practitioners with a structured foundation for advancing model merging.

Recently, a new Continual Learning (CL) paradigm was presented to control catastrophic forgetting, called Interval Continual Learning (InterContiNet), which relies on enforcing interval constraints on the neural network parameter space. Unfortunately, InterContiNet training is challenging due to the high dimensionality of the weight space, making intervals difficult to manage. To address this issue, we introduce HyperInterval, a technique that employs interval arithmetic within the embedding space and utilizes a hypernetwork to map these intervals to the target network parameter space. We train interval embeddings for consecutive tasks and train a hypernetwork to transform these embeddings into weights of the target network. An embedding for a given task is trained along with the hypernetwork, preserving the response of the target network for the previous task embeddings. Interval arithmetic works with a more manageable, lower-dimensional embedding space rather than directly preparing intervals in a high-dimensional weight space. Our model allows faster and more efficient training. Furthermore, HyperInterval maintains the guarantee of not forgetting. At the end of training, we can choose one universal embedding to produce a single network dedicated to all tasks. In such a framework, hypernetwork is used only for training and can be seen as a meta-trainer. HyperInterval obtains significantly better results than InterContiNet and gives SOTA results on several benchmarks.

Robust ASR under domain shift is crucial because real-world systems encounter unseen accents and domains with limited labeled data. Although pseudo-labeling offers a practical workaround, it often introduces systematic, accent-specific errors that filtering fails to fix. We ask: How can we correct these recurring biases without target ground truth? We propose a simple parameter-space correction: in a source domain containing both real and pseudo-labeled data, two ASR models are fine-tuned from the same initialization, one on ground-truth labels and the other on pseudo-labels, and their weight difference forms a correction vector that captures pseudo-label biases. When applied to a pseudo-labeled target model, this vector enhances recognition, achieving up to a 35% relative Word Error Rate (WER) reduction on AfriSpeech-200 across ten African accents with the Whisper tiny model.

Recent advances in music foundation models have improved audio representation learning, yet their effectiveness across diverse musical traditions remains limited. We introduce CultureMERT-95M, a multi-culturally adapted foundation model developed to enhance cross-cultural music representation learning and understanding. To achieve this, we propose a two-stage continual pre-training strategy that integrates learning rate re-warming and re-decaying, enabling stable adaptation even with limited computational resources. Training on a 650-hour multi-cultural data mix, comprising Greek, Turkish, and Indian music traditions, results in an average improvement of 4.9% in ROC-AUC and AP across diverse non-Western music auto-tagging tasks, surpassing prior state-of-the-art, with minimal forgetting on Western-centric benchmarks. We further investigate task arithmetic, an alternative approach to multi-cultural adaptation that merges single-culture adapted models in the weight space. Task arithmetic performs on par with our multi-culturally trained model on non-Western auto-tagging tasks and shows no regression on Western datasets. Cross-cultural evaluation reveals that single-culture models transfer with varying effectiveness across musical traditions, whereas the multi-culturally adapted model achieves the best overall performance. To support research on world music representation learning, we publicly release CultureMERT-95M and CultureMERT-TA-95M, fostering the development of more culturally aware music foundation models.

In-Context Learning (ICL) emerges as a key feature for Large Language Models (LLMs), allowing them to adapt to new tasks by leveraging task-specific examples without updating model parameters. However, ICL faces challenges with increasing numbers of examples due to performance degradation and quadratic computational costs. In this paper, we propose Logit Arithmetic Reweighting Approach (LARA), a novel framework that enhances ICL by using logit-based ensembling of multiple demonstrations. Our approach divides long input demonstrations into parallelizable shorter inputs to significantly reduce memory requirements, and then effectively aggregate the information by reweighting logits of each group via a non-gradient optimization approach. We further introduce Binary LARA (B-LARA), a variant that constrains weights to binary values to simplify the search space and reduces memory usage by filtering out less informative demonstration groups. Experiments on BBH and MMLU demonstrate that LARA and B-LARA outperform all baseline methods in both accuracy and memory efficiency. We also conduct extensive analysis to show that LARA generalizes well to scenarios of varying numbers of examples from limited to many-shot demonstrations.

Merging finetuned Large Language Models (LLMs) has become increasingly important for integrating diverse capabilities into a single unified model. However, prevailing model merging methods rely on linear arithmetic in Euclidean space, which often destroys the intrinsic geometric properties of pretrained weights, such as hyperspherical energy. To address this, we propose Orthogonal Model Merging (OrthoMerge), a method that performs merging operations on the Riemannian manifold formed by the orthogonal group to preserve the geometric structure of the model's weights. By mapping task-specific orthogonal matrices learned by Orthogonal Finetuning (OFT) to the Lie algebra, OrthoMerge enables a principled yet efficient integration that takes into account both the direction and intensity of adaptations. In addition to directly leveraging orthogonal matrices obtained by OFT, we further extend this approach to general models finetuned with non-OFT methods (i.e., low-rank finetuning, full finetuning) via an Orthogonal-Residual Decoupling strategy. This technique extracts the orthogonal components of expert models by solving the orthogonal Procrustes problem, which are then merged on the manifold of the orthogonal group, while the remaining linear residuals are processed through standard additive merging. Extensive empirical results demonstrate the effectiveness of OrthoMerge in mitigating catastrophic forgetting and maintaining model performance across diverse tasks.

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

Learning in weight spaces, where neural networks process the weights of other deep neural networks, has emerged as a promising research direction with applications in various fields, from analyzing and editing neural fields and implicit neural representations, to network pruning and quantization. Recent works designed architectures for effective learning in that space, which takes into account its unique, permutation-equivariant, structure. Unfortunately, so far these architectures suffer from severe overfitting and were shown to benefit from large datasets. This poses a significant challenge because generating data for this learning setup is laborious and time-consuming since each data sample is a full set of network weights that has to be trained. In this paper, we address this difficulty by investigating data augmentations for weight spaces, a set of techniques that enable generating new data examples on the fly without having to train additional input weight space elements. We first review several recently proposed data augmentation schemes %that were proposed recently and divide them into categories. We then introduce a novel augmentation scheme based on the Mixup method. We evaluate the performance of these techniques on existing benchmarks as well as new benchmarks we generate, which can be valuable for future studies.

Weight-Space Programming (WSP) enables reasoning agents to directly inspect and modify neural network parameters, offering an alternative to gradient-based model adaptation. However, its capabilities, cross-task generalization, and practical limitations remain insufficiently understood. We systematically investigate WSP through 16 editing trajectories across seven benchmarks and two model scales (Qwen3.5-0.8B and Qwen3.5-9B). Our findings reveal three key insights. First, WSP can achieve substantial performance gains through highly localized interventions. On full held-out evaluations, accuracy improves by up to 29.45, 4.42, 36.72, and 17.11 percentage points on MATH-500, GPQA Diamond, IFEval, and SciBench, respectively. Notably, modifying only 1,023 parameters yields a 29.45-point improvement. Second, WSP exhibits promising but uneven cross-task generalization. A Math-edited checkpoint improves GPQA Diamond, IFEval, and SciBench by 3.03, 12.18, and 20.10 points, respectively, without further editing, while leaving SciCode performance unchanged. Third, effective editing remains challenging and inconsistent. Outcomes vary substantially across trajectories, and six of the 16 trajectories retain the original weights when candidate improvements cannot be reliably validated. Together, these results highlight WSP's potential for highly localized model adaptation while revealing important challenges in intervention discovery, reliability, and cross-task transfer. Homepage: https://wsp-scope.github.io/ GitHub: https://github.com/WSP-Scope/WSP-Scope-Project

A challenging problem in many modern machine learning tasks is to process weight-space features, i.e., to transform or extract information from the weights and gradients of a neural network. Recent works have developed promising weight-space models that are equivariant to the permutation symmetries of simple feedforward networks. However, they are not applicable to general architectures, since the permutation symmetries of a weight space can be complicated by recurrence or residual connections. This work proposes an algorithm that automatically constructs permutation equivariant models, which we refer to as universal neural functionals (UNFs), for any weight space. Among other applications, we demonstrate how UNFs can be substituted into existing learned optimizer designs, and find promising improvements over prior methods when optimizing small image classifiers and language models. Our results suggest that learned optimizers can benefit from considering the (symmetry) structure of the weight space they optimize. We open-source our library for constructing UNFs at https://github.com/AllanYangZhou/universal_neural_functional.

Metric space magnitude, an active field of research in algebraic topology, is a scalar quantity that summarizes the effective number of distinct points that live in a general metric space. The {\em weighting vector} is a closely-related concept that captures, in a nontrivial way, much of the underlying geometry of the original metric space. Recent work has demonstrated that when the metric space is Euclidean, the weighting vector serves as an effective tool for boundary detection. We recast this result and show the weighting vector may be viewed as a solution to a kernelized SVM. As one consequence, we apply this new insight to the task of outlier detection, and we demonstrate performance that is competitive or exceeds performance of state-of-the-art techniques on benchmark data sets. Under mild assumptions, we show the weighting vector, which has computational cost of matrix inversion, can be efficiently approximated in linear time. We show how nearest neighbor methods can approximate solutions to the minimization problems defined by SVMs.

Metric space magnitude, an active subject of research in algebraic topology, originally arose in the context of biology, where it was used to represent the effective number of distinct species in an environment. In a more general setting, the magnitude of a metric space is a real number that aims to quantify the effective number of distinct points in the space. The contribution of each point to a metric space's global magnitude, which is encoded by the {\em weighting vector}, captures much of the underlying geometry of the original metric space. Surprisingly, when the metric space is Euclidean, the weighting vector also serves as an effective tool for boundary detection. This allows the weighting vector to serve as the foundation of novel algorithms for classic machine learning tasks such as classification, outlier detection and active learning. We demonstrate, using experiments and comparisons on classic benchmark datasets, the promise of the proposed magnitude and weighting vector-based approaches.

Using the weights of trained Neural Network (NN) models as data modality has recently gained traction as a research field - dubbed Weight Space Learning (WSL). Multiple recent works propose WSL methods to analyze models, evaluate methods, or synthesize weights. Weight space learning methods require populations of trained models as datasets for development and evaluation. However, existing collections of models - called `model zoos' - are unstructured or follow a rudimentary definition of diversity. In parallel, work rooted in statistical physics has identified phases and phase transitions in NN models. Models are homogeneous within the same phase but qualitatively differ from one phase to another. We combine the idea of `model zoos' with phase information to create a controlled notion of diversity in populations. We introduce 12 large-scale zoos that systematically cover known phases and vary over model architecture, size, and datasets. These datasets cover different modalities, such as computer vision, natural language processing, and scientific ML. For every model, we compute loss landscape metrics and validate full coverage of the phases. With this dataset, we provide the community with a resource with a wide range of potential applications for WSL and beyond. Evidence suggests the loss landscape phase plays a role in applications such as model training, analysis, or sparsification. We demonstrate this in an exploratory study of the downstream methods like transfer learning or model weights averaging.

Weight-space programming, introduced in recent MIT work by Takehi et al., enables a frontier LLM agent to read a network's weights and write a repair. Their GPT-6 Astra agent corrected the reasoning loops of Qwen3.5-0.8B without gradients, using 158 agent turns, 89.7 million tokens, and 50 forward-pass requests. Our method, Tree-Ising, casts this repair as a sparse discrete optimization problem over the model's own BF16 weights. Forward statistics define an objective that weakens a loop direction, strengthens a concluding direction, and preserves the layer's output on productive tokens. Within a state-tree representation, where lossless BF16 storage acts as a special case, the admissible edits are sign flips and exponent shifts that preserve all shared group exponents and mantissas. We decompose this objective exactly into independent QUBO (Ising) problems, one per output row, which we solve on a GPU using simulated annealing or quantum-inspired simulated bifurcation. On the agent's two layers, Tree-Ising achieves 54.6% on MATH-500, outperforming the agent's edit (51.7%) under the same evaluation protocol, while the base model scores 7.9% as reported. The simulated bifurcation solver yields a statistically indistinguishable repair at 55.1%. Both solvers outperform the agent edit across all five math benchmarks, with statistically significant improvements on MATH-500 and GSM8K after correcting for multiple comparisons. Our approach modifies 13% to 17% fewer weights and reduces BF16 bit flips by a factor of 2.2 to 2.3. It requires only four forward-pass requests and zero LLM tokens, with the annealing recipe costing about $0.90 in GPU time. On Qwen3.5-2B, with layers and dose selected using only the base model, the same recipe increases greedy GSM8K accuracy from 21.6% to 69.4%. As a proof of concept, we execute an eight-variable repair subproblem using QAOA on an IBM Heron processor.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

Large neural networks are increasingly deployed with 1-4 bit weights and 16- or 32-bit activations, yet their dot products still use one multiplier per element. We show that a length-n dot product with k-bit weight codes can be computed with n+2^k additions and at most 2^k decodes and 2^k multiplications. Each weight code is used directly as an address offset that selects one of 2^k accumulators; each activation is added into the selected accumulator; and the accumulated sums are decoded and multiplied only once, at the end, in a fused decode-and-multiply step. The decoded weight vector is never formed. The refactoring is exact for integer arithmetic, works with integer or floating-point operands and with arbitrary, including nonlinear, decoders, and extends to group-wise scales and to matrix and tensor products. The multiplier count falls by n/2^k, while total arithmetic energy improves by a smaller, bounded factor that we quantify with a cost model. We describe a scatter-accumulate processing element (SA-PE) that replaces the MAC unit in a weight-stationary systolic array, with a load/stream/reduce/drain schedule in which a small, time-shareable multiplier runs only 2^k times per tile. Because decoded weights are not stored, local weight memory shrinks by up to w_d/k for w_d-bit decoded weights. An RTL implementation synthesized for FPGA and compared with a matched MAC array shows that an unshared SA-PE is larger than a MAC PE, and that the saving appears when the reduction multiplier is shared across PEs. Software models confirm exact integer equivalence and floating-point error no worse than the conventional pipeline. We do not claim the underlying identity is new, and we discuss related work on weight sharing and lookup-based kernels.

Weight-space programming allows language models to read a trained neural network's weights and edit them. This opens the door to ``training'' by reasoning rather than through gradient descent. Prior work has demonstrated its functionality and promise for game-playing networks and fixing specific failures in language models, even extending to diffusion models. In this work, we present its applicability to models trained on biosignal data, specifically EEG and ECG data. We show that frontier models are not only able to debug specific faults planted in biosignal-based models without being told what these faults are, but are even capable of writing all the weights of a network from scratch, given only the architecture, the task, and a minimal set of training subjects. In particular, in one set of experiments, a language model was able to match standard gradient-descent training using 6times fewer training subjects, through reasoning alone and without any gradient descent step. Networks written by Claude Opus 5.5 also scored higher than classic hand-crafted EEG pipelines (alpha band power, and common spatial patterns with linear discriminant analysis) when built from the same 5 subjects. We also highlight some of the limitations, which include the inability of language models to improve already well-trained classification models, and their incapacity to reliably recognize, from the weights alone, the task a network was trained on. Overall, our results indicate that weight-space programming can be useful for small networks, such as those deployed on edge devices and wearables, but the concept still requires further investigation before it can be used in practice.

We investigate the space of weights spanned by a large collection of customized diffusion models. We populate this space by creating a dataset of over 60,000 models, each of which is a base model fine-tuned to insert a different person's visual identity. We model the underlying manifold of these weights as a subspace, which we term weights2weights. We demonstrate three immediate applications of this space -- sampling, editing, and inversion. First, as each point in the space corresponds to an identity, sampling a set of weights from it results in a model encoding a novel identity. Next, we find linear directions in this space corresponding to semantic edits of the identity (e.g., adding a beard). These edits persist in appearance across generated samples. Finally, we show that inverting a single image into this space reconstructs a realistic identity, even if the input image is out of distribution (e.g., a painting). Our results indicate that the weight space of fine-tuned diffusion models behaves as an interpretable latent space of identities.

Task arithmetic refers to editing the pre-trained model by adding a weighted sum of task vectors, each of which is the weight update from the pre-trained model to fine-tuned models for certain tasks. This approach recently gained attention as a computationally efficient inference method for model editing, e.g., multi-task learning, forgetting, and out-of-domain generalization capabilities. However, the theoretical understanding of why task vectors can execute various conceptual operations remains limited, due to the highly non-convexity of training Transformer-based models. To the best of our knowledge, this paper provides the first theoretical characterization of the generalization guarantees of task vector methods on nonlinear Transformers. We consider a conceptual learning setting, where each task is a binary classification problem based on a discriminative pattern. We theoretically prove the effectiveness of task addition in simultaneously learning a set of irrelevant or aligned tasks, as well as the success of task negation in unlearning one task from irrelevant or contradictory tasks. Moreover, we prove the proper selection of linear coefficients for task arithmetic to achieve guaranteed generalization to out-of-domain tasks. All of our theoretical results hold for both dense-weight parameters and their low-rank approximations. Although established in a conceptual setting, our theoretical findings were validated on a practical machine unlearning task using the large language model Phi-1.5 (1.3B).

Model merging combines multiple fine-tuned models without additional training, and arithmetic methods have emerged as a simple yet effective way to merge models without data. However, existing arithmetic methods treat task vectors in parameter space and apply a single element-wise rule to directions that can differ in mergeability. In a shared directional basis, we find a consistent two-regime structure: a low-dispersion common bulk shared across tasks and a high-dispersion task-dominated tail that carries disproportionate energy. These findings motivate Common--Unique Subspace Decomposition (CUSD), a data-free pre-merging method that separates task vectors into common and unique directional components, merges the common components with a downstream arithmetic operator, and retains unique residuals separately. Experiments across 11 base models, four model families, and four arithmetic-based merging operators show that CUSD improves 42 of 44 model--operator combinations and achieves up to 22.5% improvement over the corresponding baselines. Our code is publicly available at https://github.com/stwang666/CUSD

A challenge to Weight Space Learning as it expands from task specific and vision models to Large Language Models (LLMs) is the absence of a well curated and analyzed LLM population. To address this, we train such a model zoo for application to Weight Space Generation by systematically varying the data mixture composition between web text, books, mathematics, coding, and French across populations of 51M, 124M, and 355M parameter GPT-2 Models, each tier with count N = 100. We analyze these populations by cosine similarity, functional diversity, Leave-One-Mixture-Out modeling, and variance decomposition. We show that the zoo contains diverse models whose variation is structured by training mixture composition. We embed the training mixture composition to condition a Flow Matching model of complete pretrained LLM weights in Gram-PCA compressed weight space. By demonstrating zero-shot generalization to held out mixtures, measuring Continuous Ranked Probability Score skill, as well as finding minimal changes in downstream test-set perplexity and general capability benchmark scores, we show that our generator generalizes on the simplex of data mixtures to produce models that perform similarly to those in the zoo ex nihilo. Finally, we produce all source code and artifacts to support the continued expansion of Weight Space Learning and Generation into the LLM domain.

Fast matrix multiplication algorithms keep the product fixed and search for a cheaper way to evaluate it. We instead ask whether a Transformer's learned projections can use a different, cheaper product altogether. Building on an associative-algebra construction that replaces ordinary matrix multiplication with a sparser interaction table over the same weight blocks, we construct a family with quadratic arithmetic in the matrix dimension when the physical block size remains fixed, and derive finite-shape constraints for GPU execution. The construction is provably optimal for its bilinear rank by the Alder--Strassen bound and can be realized as row-typed rectangular projections compatible with causal masking and KV-cached decoding. We provide an empirical test of this approach by training two approximately 110M-parameter decoder-only Transformer LMs from the same recipe and 12.3B-token budget, differing only in their feed-forward layer: one uses ordinary dense matrix multiplication and the other uses the associative-algebra product. Across four prompt domains, the algebraic model achieves a 6.2--7.8\% increase in end-to-end generation throughput, while obtaining lower scores on all three reported downstream metrics. We treat these results as a feasibility and trainability check for the proposed approach at small scale, leaving further investigation to future work.

Large language models store all learned knowledge in a single, fixed weight vector. Teaching a model new capabilities requires modifying those same weights, inevitably degrading previously acquired knowledge. This fundamental limitation, known as catastrophic forgetting, has resisted principled solutions for decades. Existing approaches treat weights as immutable artifacts that must be protected through techniques like regularization heuristics, replay buffers, or isolated adapter modules. The problem is none of these provide a structural guarantee against forgetting. In this work, we propose Non-Interfering Weight Fields (NIWF), a framework that replaces the fixed weight paradigm with a learned function that generates weight configurations on demand from a continuous capability coordinate space. After training on a task, we commit the occupied coordinate region by snapshotting the fields outputs on anchor points to enforce a functional lock during all future training. We validate NIWF on sequential instructionfollowing and code generation tasks using Mistral-7B, demonstrating zero forgetting on committed tasks with competitive perplexity on new tasks. The framework introduces the notion of software-like versioning for neural network intelligence, where capabilities can be committed, extended, composed, and rolled back without retraining.

In this work, we study how well the learned weights of a neural network utilize the space available to them. This notion is related to capacity, but additionally incorporates the interaction of the network architecture with the dataset. Most learned weights appear to be full rank, and are therefore not amenable to low rank decomposition. This deceptively implies that the weights are utilizing the entire space available to them. We propose a simple data-driven transformation that projects the weights onto the subspace where the data and the weight interact. This preserves the functional mapping of the layer and reveals its low rank structure. In our findings, we conclude that most models utilize a fraction of the available space. For instance, for ViTB-16 and ViTL-16 trained on ImageNet, the mean layer utilization is 35% and 20% respectively. Our transformation results in reducing the parameters to 50% and 25% respectively, while resulting in less than 0.2% accuracy drop after fine-tuning. We also show that self-supervised pre-training drives this utilization up to 70%, justifying its suitability for downstream tasks.

Ternary weight quantization (e.g., BitNet b1.58) offers a promising path to mitigate the memory bandwidth bottleneck in Large Language Model (LLM) inference. However, conventional compute platforms lack native support for ternary-weight arithmetic, often relying on inefficient dequantization. Lookup table (LUT)-based hardware architectures provide an effective alternative by replacing multiplications with conditional additions, but their design space remains largely unexplored. Existing designs rely on heuristic parameter selection, lacking a systematic understanding of the architectural trade-offs. This work addresses this gap by formalizing the design space of ternary LUT-based accelerators and presenting an open-source hardware generator coupled with an analytical cost model, validated against synthesis in TSMC 16nm technology. By spanning the full architectural space, this framework not only enables rapid design space exploration but also establishes a common footing for fair cross-design evaluation, which was previously hindered by inconsistent instantiations across published accelerators. Using this framework, we challenge several assumptions and design choices in recent literature. We demonstrate that the optimal architecture is fundamentally governed by the activation data type: while LUT-based reuse offers significant gains for high-cost arithmetic (e.g., FP16), it yields diminishing returns for small integer types. Furthermore, we show that maximizing core size consistently improves area density compared to highly tiled approaches. Our optimized designs achieve a 2.2x area reduction compared to multiplier-based baselines. Moreover, by benchmarking state-of-the-art implementations against our model, we reveal that correcting suboptimal parameters yields up to a 1.2x area improvement.

Modeling stochastic dynamics from discrete observations is a key interdisciplinary challenge. Existing methods often fail to estimate the continuous evolution of probability densities from trajectories or face the curse of dimensionality. To address these limitations, we presents a novel paradigm: modeling dynamics directly in the weight space of a neural network by projecting the evolving probability distribution. We first theoretically establish the connection between dynamic optimal transport in measure space and an equivalent energy functional in weight space. Subsequently, we design WeightFlow, which constructs the neural network weights into a graph and learns its evolution via a graph controlled differential equation. Experiments on interdisciplinary datasets demonstrate that WeightFlow improves performance by an average of 43.02\% over state-of-the-art methods, providing an effective and scalable solution for modeling high-dimensional stochastic dynamics.

Hyperdimensional computing (HDC) is a biologically-inspired framework which represents symbols with high-dimensional vectors, and uses vector operations to manipulate them. The ensemble of a particular vector space and a prescribed set of vector operations (including one addition-like for "bundling" and one outer-product-like for "binding") form a *vector symbolic architecture* (VSA). While VSAs have been employed in numerous applications and have been studied empirically, many theoretical questions about VSAs remain open. We analyze the *representation capacities* of four common VSAs: MAP-I, MAP-B, and two VSAs based on sparse binary vectors. "Representation capacity' here refers to bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks, such as testing for set membership i in S and estimating set intersection sizes |X cap Y| for two sets of symbols X and Y, to a given degree of accuracy. We also analyze the ability of a novel variant of a Hopfield network (a simple model of associative memory) to perform some of the same tasks that are typically asked of VSAs. In addition to providing new bounds on VSA capacities, our analyses establish and leverage connections between VSAs, "sketching" (dimensionality reduction) algorithms, and Bloom filters.

Deep learning models like Transformers and Convolutional Neural Networks (CNNs) have revolutionized various domains, but their parameter-intensive nature hampers deployment in resource-constrained settings. In this paper, we introduce a novel concept utilizes column space and row space of weight matrices, which allows for a substantial reduction in model parameters without compromising performance. Leveraging this paradigm, we achieve parameter-efficient deep learning models.. Our approach applies to both Bottleneck and Attention layers, effectively halving the parameters while incurring only minor performance degradation. Extensive experiments conducted on the ImageNet dataset with ViT and ResNet50 demonstrate the effectiveness of our method, showcasing competitive performance when compared to traditional models. This approach not only addresses the pressing demand for parameter efficient deep learning solutions but also holds great promise for practical deployment in real-world scenarios.

This work explores the in-context learning capabilities of State Space Models (SSMs) and presents, to the best of our knowledge, the first theoretical explanation of a possible underlying mechanism. We introduce a novel weight construction for SSMs, enabling them to predict the next state of any dynamical system after observing previous states without parameter fine-tuning. This is accomplished by extending the HiPPO framework to demonstrate that continuous SSMs can approximate the derivative of any input signal. Specifically, we find an explicit weight construction for continuous SSMs and provide an asymptotic error bound on the derivative approximation. The discretization of this continuous SSM subsequently yields a discrete SSM that predicts the next state. Finally, we demonstrate the effectiveness of our parameterization empirically. This work should be an initial step toward understanding how sequence models based on SSMs learn in context.

Requiring 10^{13}-10^{15} FLOPs to calculate one 8 bit weight in an LLM during pretraining is extremely expensive and seems inefficient. To better leverage the huge investments made into pretrained models, we develop the new "Assembly-of-Experts" (AoE) construction method to create capable child variants of existing Mixture-of-Experts parent models in linear time. Model weight tensors get interpolated individually, allowing to enhance or suppress semantic features of the parents. Varying the proportion of weights taken from the parent models, we observe some properties of the AoE child model changing gradually, while other behavioral traits emerge with a sharp transition. Surprisingly, nearly every generated model is functional and capable, which makes searching the model space straightforward. We construct the DeepSeek R1T "Chimera", a 671B open-weights hybrid model combining DeepSeek's V3-0324 and R1 model variants. The child inherits only the routed expert tensors of R1, but still achieves about R1-level intelligence. At the same time, it uses about 40\% fewer output tokens, close to V3 speed. Constructed without any fine-tuning or distillation, the Chimera exhibits surprisingly compact, orderly reasoning compared to its parent models.

Mixture of Experts (MoE) models scale capacity but often suffer from representation collapse and gradient instability. We propose Dynamic Subspace Composition (DSC), a framework that approximates context-dependent weights via a state-dependent, sparse expansion of a shared basis bank. Formally, DSC models the weight update as a residual trajectory within a Star- Shaped Domain, employing a Magnitude-Gated Simplex Interpolation to ensure continuity at the identity. Unlike standard Mixture-of-LoRAs, which incurs O(M rd) parameter complexity by retrieving independent rank-r matrices, DSC constructs a compositional rank-K approximation from decoupled unit-norm basis vectors. This reduces parameter complexity to O(M d) and memory traffic to O(Kd), while Frame-Theoretic regularization and spectral constraints provide rigorous worst-case bounds on the dynamic update. The code is available at https://github. com/VladimerKhasia/DSC

We develop tools for explicitly constructing categories enriched over generating data and that compose via ordinary scalar and matrix arithmetic arithmetic operations. We characterize meaningful size maps, weightings, and magnitude that reveal features analogous to outliers that these same notions have previously been shown to reveal in the context of metric spaces. Throughout, we provide examples of such "outlier detection" relevant to the analysis of computer programs, neural networks, cyber-physical systems, and networks of communications channels.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Task arithmetic provides an efficient, training-free way to edit pre-trained models, yet lacks a fundamental theoretical explanation for its success. The existing concept of ``weight disentanglement" describes the ideal outcome of non-interfering task composition but does not reveal its underlying cause. Crucially, what intrinsic properties of the pre-trained model (θ_0) or the task vectors (τ_t) enable this disentanglement remains underexplored. In this paper, we introduce Task-Feature Specialization (TFS), a model's ability to allocate distinct internal features to different tasks, as the fundamental principle. We first prove that TFS is a sufficient condition for weight disentanglement. More importantly, we find that TFS also gives rise to an observable geometric consequence: weight vector orthogonality. This positions TFS as the common cause for both the desired functional outcome (disentanglement) and a measurable geometric property (orthogonality). This relationship provides the key insight for our method: since the abstract TFS property is intractable to enforce directly, we can instead promote weight disentanglement by shaping its concrete geometric consequence, orthogonality. Therefore, we propose OrthoReg, a simple and effective regularization method that actively enforces an internal orthogonal structure on weight updates (ΔW) that constitute τ_t during fine-tuning. And we theoretically prove that OrthoReg promotes disentanglement. Extensive experiments demonstrate that OrthoReg consistently and significantly enhances the performance of various task arithmetic methods. Code is available at https://github.com/RL-MIND/OrthoReg{https://github.com/RL-MIND/OrthoReg}.

Designing machine learning architectures for processing neural networks in their raw weight matrix form is a newly introduced research direction. Unfortunately, the unique symmetry structure of deep weight spaces makes this design very challenging. If successful, such architectures would be capable of performing a wide range of intriguing tasks, from adapting a pre-trained network to a new domain to editing objects represented as functions (INRs or NeRFs). As a first step towards this goal, we present here a novel network architecture for learning in deep weight spaces. It takes as input a concatenation of weights and biases of a pre-trained MLP and processes it using a composition of layers that are equivariant to the natural permutation symmetry of the MLP's weights: Changing the order of neurons in intermediate layers of the MLP does not affect the function it represents. We provide a full characterization of all affine equivariant and invariant layers for these symmetries and show how these layers can be implemented using three basic operations: pooling, broadcasting, and fully connected layers applied to the input in an appropriate manner. We demonstrate the effectiveness of our architecture and its advantages over natural baselines in a variety of learning tasks.

Many real-world machine learning problems involve inherently hierarchical data, yet traditional approaches rely on Euclidean metrics that fail to capture the discrete, branching nature of hierarchical relationships. We present a theoretical foundation for machine learning in p-adic metric spaces, which naturally respect hierarchical structure. Our main result proves that an n-dimensional plane minimizing the p-adic sum of distances to points in a dataset must pass through at least n + 1 of those points -- a striking contrast to Euclidean regression that highlights how p-adic metrics better align with the discrete nature of hierarchical data. As a corollary, a polynomial of degree n constructed to minimise the p-adic sum of residuals will pass through at least n + 1 points. As a further corollary, a polynomial of degree n approximating a higher degree polynomial at a finite number of points will yield a difference polynomial that has distinct rational roots. We demonstrate the practical significance of this result through two applications in natural language processing: analyzing hierarchical taxonomies and modeling grammatical morphology. These results suggest that p-adic metrics may be fundamental to properly handling hierarchical data structures in machine learning. In hierarchical data, interpolation between points often makes less sense than selecting actual observed points as representatives.

Task arithmetic, representing downstream tasks through linear operations on task vectors, has emerged as a simple yet powerful paradigm for transferring knowledge across diverse settings. However, maintaining a large collection of task vectors introduces scalability challenges in both storage and computation. We propose Task Vector Bases, a framework compressing T task vectors into M < T basis vectors while preserving the functionality of task arithmetic. By representing each task vector as a structured linear combination of basis atoms, our approach supports standard operations such as addition, negation, as well as more advanced arithmetic ones. The framework is orthogonal to other efficiency-oriented improvements in task arithmetic and can be used in combination with them. We provide theoretical analysis showing that basis compression retains addition generalization guarantees and enables principled unlearning, with error bounds depending on reconstruction quality. Empirically, our proposed basis construction methods consistently outperform heuristic basis construction baselines and, in some cases, even surpass the performance of full task vector collections across diverse downstream applications while reducing storage and computational requirements. The code is available at https://github.com/uiuctml/TaskVectorBasis.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

A language model can repair a robot policy's lexical instruction grounding by editing just 65 token-embedding rows. We bring weight-space programming to robot foundation policies: GPT-6 Astra reads weights and uses a CPU workspace and budgeted simulator feedback, without gradient updates or new demonstrations. Three independent sessions on π_{0.5} all improve success on 765 held-out paraphrases; the best raises success from 86.3\% to 92.0\% (48 repairs, four regressions; exact paired p=1.31times10^{-10}). Among the 335 instructions containing edited tokens, success rises from 74.3\% to 87.5\%; the other 430 episodes have zero outcome changes. Matched random-direction edits do not help. Full replacement using the editor's own lexicon explains much of the gain but produces more regressions; its direct difference from the editor is not significant at 0.05. Across 13 further robot sessions, the editor declines 12 candidates; the sole retained edit does not show a significant held-out gain. Separately, benchmark metadata lowers FastWAM success from 70.0\% to 48.6\%, while the evaluated VLA's aggregate rate is unchanged. The evidence supports precise lexical repair, not general acquisition of missing robot capabilities.

Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence. Due to the vast number of tasks and high costs, adjusting all parameters becomes unfeasible. To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner, such as through low-rank adjustments. Yet, almost all of these methods focus on linear weights, neglecting the intricacies of parameter spaces in higher dimensions like 4D. Alternatively, some methods can be adapted for high-dimensional parameter space by compressing changes in the original space into two dimensions and then employing low-rank matrix decomposition. However, these approaches destructs the structural integrity of the involved high-dimensional spaces. To tackle the diversity of dimensional spaces across different foundation models and provide a more precise representation of the changes within these spaces, this paper introduces a generalized parameter-efficient fine-tuning framework, FLoRA, designed for various dimensional parameter space. Specifically, utilizing Tucker decomposition, FLoRA asserts that changes in each dimensional parameter space are based on a low-rank core space which maintains the consistent topological structure with the original space. It then models the changes through this core space alongside corresponding weights to reconstruct alterations in the original space. FLoRA effectively preserves the structural integrity of the change of original N-dimensional parameter space, meanwhile decomposes it via low-rank tensor decomposition. Extensive experiments on computer vision, natural language processing and multi-modal tasks validate FLoRA's effectiveness. Codes are available at https://github.com/SJTU-DeepVisionLab/FLoRA.

Artificial Intelligence and Machine learning have been widely used in various fields of mathematical computing, physical modeling, computational science, communication science, and stochastic analysis. Approaches based on Deep Artificial Neural Networks (DANN) are very popular in our days. Depending on the learning task, the exact form of DANNs is determined via their multi-layer architecture, activation functions and the so-called loss function. However, for a majority of deep learning approaches based on DANNs, the kernel structure of neural signal processing remains the same, where the node response is encoded as a linear superposition of neural activity, while the non-linearity is triggered by the activation functions. In the current paper, we suggest to analyze the neural signal processing in DANNs from the point of view of homogeneous chaos theory as known from polynomial chaos expansion (PCE). From the PCE perspective, the (linear) response on each node of a DANN could be seen as a 1^{st} degree multi-variate polynomial of single neurons from the previous layer, i.e. linear weighted sum of monomials. From this point of view, the conventional DANN structure relies implicitly (but erroneously) on a Gaussian distribution of neural signals. Additionally, this view revels that by design DANNs do not necessarily fulfill any orthogonality or orthonormality condition for a majority of data-driven applications. Therefore, the prevailing handling of neural signals in DANNs could lead to redundant representation as any neural signal could contain some partial information from other neural signals. To tackle that challenge, we suggest to employ the data-driven generalization of PCE theory known as arbitrary polynomial chaos (aPC) to construct a corresponding multi-variate orthonormal representations on each node of a DANN to obtain Deep arbitrary polynomial chaos neural networks.

We present a model inspired by the Global Workspace Theory that integrates specialized modules to perform a sequential reasoning task. A controller selectively routes information between modules through the workspace using a gating mechanism. This approach allows the model to chain operations by iteratively broadcasting information between specialized domains, mimicking System-2 reasoning. We evaluate the model's performance on a simple addition task, where two addends must be summed. The task can be solved by routing information sequentially through an Input module, an Increment module (multiple times), and finally an Output module. We consider two implementations of this system with increasing complexity. First, using hand-designed modules operating on one-hot digit representations, the controller (a LSTM recurrent network) learns to select the appropriate modules (input, increment, output) in the appropriate sequence. Second, we replace the hand-designed modules with learned representation modules for MNIST images and an increment module trained on the task objectives; here again, the controller learns the appropriate sequential module selection to solve the task. Finally, we show that the Global Workspace model, while having fewer parameters, outperforms LSTMs and Transformers when tested on unseen addition operations (both interpolations and extrapolations of addition operations seen during training). Our results highlight the potential of architectures inspired by the Global Workspace Theory to enhance deep learning's reasoning capabilities.

Weight sharing is a fundamental concept in neural architecture search (NAS), enabling gradient-based methods to explore cell-based architecture spaces significantly faster than traditional blackbox approaches. In parallel, weight entanglement has emerged as a technique for intricate parameter sharing among architectures within macro-level search spaces. %However, the macro structure of such spaces poses compatibility challenges for gradient-based NAS methods. %As a result, blackbox optimization methods have been commonly employed, particularly in conjunction with supernet training, to maintain search efficiency. %Due to the inherent differences in the structure of these search spaces, these Since weight-entanglement poses compatibility challenges for gradient-based NAS methods, these two paradigms have largely developed independently in parallel sub-communities. This paper aims to bridge the gap between these sub-communities by proposing a novel scheme to adapt gradient-based methods for weight-entangled spaces. This enables us to conduct an in-depth comparative assessment and analysis of the performance of gradient-based NAS in weight-entangled search spaces. Our findings reveal that this integration of weight-entanglement and gradient-based NAS brings forth the various benefits of gradient-based methods (enhanced performance, improved supernet training properties and superior any-time performance), while preserving the memory efficiency of weight-entangled spaces. The code for our work is openly accessible https://anonymous.4open.science/r/TangleNAS-527C{here}

Recent studies have shown that supervised fine-tuning of LLMs on a small number of high-quality datasets can yield strong reasoning capabilities. However, full fine-tuning (Full FT), while powerful, is computationally expensive and susceptible to overfitting and catastrophic forgetting, particularly when data is limited. Sparse fine-tuning, which previously achieved notable success by updating only a small subset of model parameters, offers a promising trade-off between efficiency and effectiveness. Yet, it has lagged behind in the LLM era due to the difficulty of identifying parameters truly critical for reasoning. In this work, we state that weights with the largest magnitude after low-rank approximation are critical weights for fine-tuning, which we call Principal Weights. Surprisingly, while magnitude-based sparse fine-tuning performs poorly as a baseline on LLM fine-tuning, it becomes highly effective after rank reduction. These insights motivate our method: Low-rank Informed Sparse Fine-Tuning (LIFT). LIFT only updates the top 5% Principal Weights throughout training and consistently achieves better performance on reasoning tasks than Full FT, while maintaining memory efficiency on par with popular parameter-efficient fine-tuning methods. In addition to strong performance on target domains such as arithmetic reasoning, LIFT also retains up to 20% more source-domain knowledge, compared to Full FT and LoRA. Our code is available at: https://github.com/zihanghliu/LIFT.

Projection maintenance is one of the core data structure tasks. Efficient data structures for projection maintenance have led to recent breakthroughs in many convex programming algorithms. In this work, we further extend this framework to the Kronecker product structure. Given a constraint matrix {sf A} and a positive semi-definite matrix Win R^{ntimes n} with a sparse eigenbasis, we consider the task of maintaining the projection in the form of {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}, where {sf B}={sf A}(Wotimes I) or {sf B}={sf A}(W^{1/2}otimes W^{1/2}). At each iteration, the weight matrix W receives a low rank change and we receive a new vector h. The goal is to maintain the projection matrix and answer the query {sf B}^top({sf B}{sf B}^top)^{-1}{sf B}h with good approximation guarantees. We design a fast dynamic data structure for this task and it is robust against an adaptive adversary. Following the beautiful and pioneering work of [Beimel, Kaplan, Mansour, Nissim, Saranurak and Stemmer, STOC'22], we use tools from differential privacy to reduce the randomness required by the data structure and further improve the running time.

This paper presents a new state-of-the-art algorithm for exact 3times3 matrix multiplication over general non-commutative rings, achieving a rank-23 scheme with only 58 scalar additions. This improves the previous best additive complexity of 60 additions without a change of basis. The result was discovered through an automated search combining ternary-restricted flip-graph exploration with greedy intersection reduction for common subexpression elimination. The resulting scheme uses only coefficients from {-1, 0, 1}, ensuring both efficiency and portability across arbitrary fields. The total scalar operation count is reduced from 83 to 81.

Weight initialization plays an important role in neural network training. Widely used initialization methods are proposed and evaluated for networks that are trained from scratch. However, the growing number of pretrained models now offers new opportunities for tackling this classical problem of weight initialization. In this work, we introduce weight selection, a method for initializing smaller models by selecting a subset of weights from a pretrained larger model. This enables the transfer of knowledge from pretrained weights to smaller models. Our experiments demonstrate that weight selection can significantly enhance the performance of small models and reduce their training time. Notably, it can also be used together with knowledge distillation. Weight selection offers a new approach to leverage the power of pretrained models in resource-constrained settings, and we hope it can be a useful tool for training small models in the large-model era. Code is available at https://github.com/OscarXZQ/weight-selection.

Foundation models are transitioning from offline predictors to deployed systems expected to operate over long time horizons. In real deployments, objectives are not fixed: domains drift, user preferences evolve, and new tasks appear after the model has shipped. This elevates continual learning and instant personalization from optional features to core architectural requirements. Yet most adaptation pipelines still follow a static weight paradigm: after training (or after any adaptation step), inference executes a single parameter vector regardless of user intent, domain, or instance-specific constraints. This treats the trained or adapted model as a single point in parameter space. In heterogeneous and continually evolving regimes, distinct objectives can induce separated feasible regions over parameters, forcing any single shared update into compromise, interference, or overspecialization. As a result, continual learning and personalization are often implemented as repeated overwriting of shared weights, risking degradation of previously learned behaviors. We propose HY-WU (Weight Unleashing), a memory-first adaptation framework that shifts adaptation pressure away from overwriting a single shared parameter point. HY-WU implements functional (operator-level) memory as a neural module: a generator that synthesizes weight updates on-the-fly from the instance condition, yielding instance-specific operators without test-time optimization.

Large neural networks spend most computation on floating point tensor multiplications. In this work, we find that a floating point multiplier can be approximated by one integer adder with high precision. We propose the linear-complexity multiplication L-Mul algorithm that approximates floating point number multiplication with integer addition operations. The new algorithm costs significantly less computation resource than 8-bit floating point multiplication but achieves higher precision. Compared to 8-bit floating point multiplications, the proposed method achieves higher precision but consumes significantly less bit-level computation. Since multiplying floating point numbers requires substantially higher energy compared to integer addition operations, applying the L-Mul operation in tensor processing hardware can potentially reduce 95% energy cost by element-wise floating point tensor multiplications and 80% energy cost of dot products. We calculated the theoretical error expectation of L-Mul, and evaluated the algorithm on a wide range of textual, visual, and symbolic tasks, including natural language understanding, structural reasoning, mathematics, and commonsense question answering. Our numerical analysis experiments agree with the theoretical error estimation, which indicates that L-Mul with 4-bit mantissa achieves comparable precision as float8_e4m3 multiplications, and L-Mul with 3-bit mantissa outperforms float8_e5m2. Evaluation results on popular benchmarks show that directly applying L-Mul to the attention mechanism is almost lossless. We further show that replacing all floating point multiplications with 3-bit mantissa L-Mul in a transformer model achieves equivalent precision as using float8_e4m3 as accumulation precision in both fine-tuning and inference.

The weights of neural networks (NNs) have recently gained prominence as a new data modality in machine learning, with applications ranging from accuracy and hyperparameter prediction to representation learning or weight generation. One approach to leverage NN weights involves training autoencoders (AEs), using contrastive and reconstruction losses. This allows such models to be applied to a wide variety of downstream tasks, and they demonstrate strong predictive performance and low reconstruction error. However, despite the low reconstruction error, these AEs reconstruct NN models with deteriorated performance compared to the original ones, limiting their usability with regard to model weight generation. In this paper, we identify a limitation of weight-space AEs, specifically highlighting that a structural loss, that uses the Euclidean distance between original and reconstructed weights, fails to capture some features critical for reconstructing high-performing models. We analyze the addition of a behavioral loss for training AEs in weight space, where we compare the output of the reconstructed model with that of the original one, given some common input. We show a strong synergy between structural and behavioral signals, leading to increased performance in all downstream tasks evaluated, in particular NN weights reconstruction and generation.

Modern artificial intelligence excels at statistical interpolation within seen manifolds but fundamentally fails at the exact reasoning required for theoretical physics and mathematics. We identify the "Float Wall" -- a catastrophic collapse of neural extrapolation at scales beyond 10^{16} -- caused by standard floating-point representation and linguistic tokenization (BPE). To resolve this, we introduce the Active Discoverer Framework, a digit-native neuro-symbolic architecture designed for invariant discovery. At its core is NumberNet, a Siamese Arithmetic Transformer that utilizes least-significant-bit (LSB) sequence encoding to achieve 0% precision loss and cosmic-scale extrapolation up to 10^{50}. To enforce physical grounding, we implement a Hamiltonian-based energy descent and Symmetry Grouping layer, ensuring the model respects Noether's theorem natively. The primary innovation is the Symbolic LaTeX Bottleneck: an active discovery loop where the model is forced to hypothesize unknown physical variables through an autoregressive LaTeX decoder. By reconciling numeric "hallucinations" with structurally valid mathematical expressions, the framework ensures that any discovered physics is parsimonious and human-interpretable. We evaluate this system against a 30-billion scale benchmark and the Universal Physics Pantheon, featuring 50 "Chaos Mode" systemic perturbations. Our results demonstrate that while traditional GBDT and LLM-based architectures collapse at cosmic scales, the Active Discoverer autonomously deduces universal constants such as the Gravitational Constant (G) with high fidelity. This framework establishes a path toward zero-hallucination artificial intelligence and truly autonomous scientific research agents.

As the issue of robustness in AI systems becomes vital, statistical learning techniques that are reliable even in presence of partly contaminated data have to be developed. Preference data, in the form of (complete) rankings in the simplest situations, are no exception and the demand for appropriate concepts and tools is all the more pressing given that technologies fed by or producing this type of data (e.g. search engines, recommending systems) are now massively deployed. However, the lack of vector space structure for the set of rankings (i.e. the symmetric group S_n) and the complex nature of statistics considered in ranking data analysis make the formulation of robustness objectives in this domain challenging. In this paper, we introduce notions of robustness, together with dedicated statistical methods, for Consensus Ranking the flagship problem in ranking data analysis, aiming at summarizing a probability distribution on S_n by a median ranking. Precisely, we propose specific extensions of the popular concept of breakdown point, tailored to consensus ranking, and address the related computational issues. Beyond the theoretical contributions, the relevance of the approach proposed is supported by an experimental study.

The weight matrix (WM) of a neural network (NN) is its program. The programs of many traditional NNs are learned through gradient descent in some error function, then remain fixed. The WM of a self-referential NN, however, can keep rapidly modifying all of itself during runtime. In principle, such NNs can meta-learn to learn, and meta-meta-learn to meta-learn to learn, and so on, in the sense of recursive self-improvement. While NN architectures potentially capable of implementing such behaviour have been proposed since the '90s, there have been few if any practical studies. Here we revisit such NNs, building upon recent successes of fast weight programmers and closely related linear Transformers. We propose a scalable self-referential WM (SRWM) that learns to use outer products and the delta update rule to modify itself. We evaluate our SRWM in supervised few-shot learning and in multi-task reinforcement learning with procedurally generated game environments. Our experiments demonstrate both practical applicability and competitive performance of the proposed SRWM. Our code is public.

Adapting a pre-trained foundation model on downstream tasks should ensure robustness against distribution shifts without the need to retrain the whole model. Although existing weight interpolation methods are simple yet effective, we argue their static nature limits downstream performance while achieving efficiency. In this work, we propose DaWin, a training-free dynamic weight interpolation method that leverages the entropy of individual models over each unlabeled test sample to assess model expertise, and compute per-sample interpolation coefficients dynamically. Unlike previous works that typically rely on additional training to learn such coefficients, our approach requires no training. Then, we propose a mixture modeling approach that greatly reduces inference overhead raised by dynamic interpolation. We validate DaWin on the large-scale visual recognition benchmarks, spanning 14 tasks across robust fine-tuning -- ImageNet and derived five distribution shift benchmarks -- and multi-task learning with eight classification tasks. Results demonstrate that DaWin achieves significant performance gain in considered settings, with minimal computational overhead. We further discuss DaWin's analytic behavior to explain its empirical success.

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

This paper introduces an efficient fine-tuning method for large pre-trained models, offering strong in-distribution (ID) and out-of-distribution (OOD) performance. Breaking away from traditional practices that need a multitude of fine-tuned models for averaging, our approach employs significantly fewer models to achieve final weights yet yield superior accuracy. Drawing from key insights in the weight space of fine-tuned weights, we uncover a strong link between the performance and proximity to the center of weight space. Based on this, we introduce a method that approximates a center-close weight using only two fine-tuned models, applicable during or after training. Our innovative layer-wise weight averaging technique surpasses state-of-the-art model methods such as Model Soup, utilizing only two fine-tuned models. This strategy can be aptly coined Model Stock, highlighting its reliance on selecting a minimal number of models to draw a more optimized-averaged model. We demonstrate the efficacy of Model Stock with fine-tuned models based upon pre-trained CLIP architectures, achieving remarkable performance on both ID and OOD tasks on the standard benchmarks, all while barely bringing extra computational demands. Our code and pre-trained models are available at https://github.com/naver-ai/model-stock.

Composition models of distributional semantics are used to construct phrase representations from the representations of their words. Composition models are typically situated on two ends of a spectrum. They either have a small number of parameters but compose all phrases in the same way, or they perform word-specific compositions at the cost of a far larger number of parameters. In this paper we propose transformation weighting (TransWeight), a composition model that consistently outperforms existing models on nominal compounds, adjective-noun phrases and adverb-adjective phrases in English, German and Dutch. TransWeight drastically reduces the number of parameters needed compared to the best model in the literature by composing similar words in the same way.

We present a memory and computation efficient ternary weight networks (TWNs) - with weights constrained to +1, 0 and -1. The Euclidian distance between full (float or double) precision weights and the ternary weights along with a scaling factor is minimized in training stage. Besides, a threshold-based ternary function is optimized to get an approximated solution which can be fast and easily computed. TWNs have shown better expressive abilities than binary precision counterparts. Meanwhile, TWNs achieve up to 16times model compression rate and need fewer multiplications compared with the float32 precision counterparts. Extensive experiments on MNIST, CIFAR-10, and ImageNet datasets show that the TWNs achieve much better result than the Binary-Weight-Networks (BWNs) and the classification performance on MNIST and CIFAR-10 is very close to the full precision networks. We also verify our method on object detection task and show that TWNs significantly outperforms BWN by more than 10\% mAP on PASCAL VOC dataset. The pytorch version of source code is available at: https://github.com/Thinklab-SJTU/twns.

Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, f:XtoY. Is it possible to identify a pair of non-standard vector spaces for which a conventionally nonlinear function is, in fact, linear? This paper introduces a method that makes such vector spaces explicit by construction. We find that if we sandwich a linear operator A between two invertible neural networks, f(x)=g_y^{-1}(A g_x(x)), then the corresponding vector spaces X and Y are induced by newly defined addition and scaling actions derived from g_x and g_y. We term this kind of architecture a Linearizer. This framework makes the entire arsenal of linear algebra, including SVD, pseudo-inverse, orthogonal projection and more, applicable to nonlinear mappings. Furthermore, we show that the composition of two Linearizers that share a neural network is also a Linearizer. We leverage this property and demonstrate that training diffusion models using our architecture makes the hundreds of sampling steps collapse into a single step. We further utilize our framework to enforce idempotency (i.e. f(f(x))=f(x)) on networks leading to a globally projective generative model and to demonstrate modular style transfer.

Large language models make weight storage and memory traffic major inference costs, motivating low-precision formats that represent each weight with only a few bits. Such formats use a scale to map floating-point values into a small codebook; NVFP4 improves local range utilization by letting every 16 E2M1 weights share an E4M3 block scale. Choosing that scale is difficult in GPTQ because quantizing one column updates those that follow, so evaluating a block independently can misestimate its final reconstruction error. Large models pose a second challenge: full-precision weights, calibration activations, and second-order state cannot all remain on one accelerator, while assigning complete layers to devices leaves each time-consuming layer solve serial. We introduce Schur Replay, a scale-selection algorithm that reproduces the GPTQ updates caused by each block scale and scores the resulting block error after accounting for compensation from unquantized columns. Separately, our execution infrastructure keeps only the active layer resident, tiers activations across device, host, and disk, retires full-precision layers after export, and distributes independent output rows across tensor-parallel ranks. Together, the algorithm and infrastructure attain 99.35% and 100.84% question-weighted recovery from BF16 across seven benchmarks on Qwen3.5-397B-A17B and Llama-3.3-70B-Instruct. On the 397B model, the infrastructure reduces measured per-layer time by 15.17times over ModelOpt and 23.14times over LLM Compressor, with lower memory used per GPU.

Large language models (LLMs) are typically pretrained in high precision but increasingly deployed with low-precision post-training quantization (PTQ). Recent studies have shown that using weight averaging during pretraining can improve PTQ performance compared with learning-rate decay, suggesting that it might provide a simple way to improve the pretraining-to-quantization transition. But the mechanism behind weight averaging remains insufficiently explained. This leads to inconsistent and fragile performance gains, thereby preventing practitioners from applying such a technique confidently. As a response, we formulate weight averaging as a trade-off between retaining training progress and improving robustness under perturbation. We further derive a continuous family of averaging kernels that unifies conventional strategies and achieves the Pareto frontier between the two competing goals. Critically, a theoretical framework for performing weight averaging under PTQ is developed. It can be shown that coarser quantization is more susceptible to perturbations, whereas finer quantization could be less affected. Thus, our results could provide unified theoretical guidance for performing weight averaging under different PTQ conditions. Experiments validate both the predicted behavior and the proposed averaging strategy. Code is available at https://github.com/MOFA-LAB/weight-averaging-for-ptq.

Deep learning optimizers are often motivated through a mix of convex and approximate second-order theory. We select three such methods -- Adam, Shampoo and Prodigy -- and argue that each method can instead be understood as a squarely first-order method without convexity assumptions. In fact, after switching off exponential moving averages, each method is equivalent to steepest descent under a particular norm. By generalizing this observation, we chart a new design space for training algorithms. Different operator norms should be assigned to different tensors based on the role that the tensor plays within the network. For example, while linear and embedding layers may have the same weight space of R^{mtimes n}, these layers play different roles and should be assigned different norms. We hope that this idea of carefully metrizing the neural architecture might lead to more stable, scalable and indeed faster training.

We present weight normalization: a reparameterization of the weight vectors in a neural network that decouples the length of those weight vectors from their direction. By reparameterizing the weights in this way we improve the conditioning of the optimization problem and we speed up convergence of stochastic gradient descent. Our reparameterization is inspired by batch normalization but does not introduce any dependencies between the examples in a minibatch. This means that our method can also be applied successfully to recurrent models such as LSTMs and to noise-sensitive applications such as deep reinforcement learning or generative models, for which batch normalization is less well suited. Although our method is much simpler, it still provides much of the speed-up of full batch normalization. In addition, the computational overhead of our method is lower, permitting more optimization steps to be taken in the same amount of time. We demonstrate the usefulness of our method on applications in supervised image recognition, generative modelling, and deep reinforcement learning.

In observational studies, balancing covariates in different treatment groups is essential to estimate treatment effects. One of the most commonly used methods for such purposes is weighting. The performance of this class of methods usually depends on strong regularity conditions for the underlying model, which might not hold in practice. In this paper, we investigate weighting methods from a functional estimation perspective and argue that the weights needed for covariate balancing could differ from those needed for treatment effects estimation under low regularity conditions. Motivated by this observation, we introduce a new framework of weighting that directly targets the treatment effects estimation. Unlike existing methods, the resulting estimator for a treatment effect under this new framework is a simple kernel-based U-statistic after applying a data-driven transformation to the observed covariates. We characterize the theoretical properties of the new estimators of treatment effects under a nonparametric setting and show that they are able to work robustly under low regularity conditions. The new framework is also applied to several numerical examples to demonstrate its practical merits.

Stacking heterogeneous vision backbones (CNNs, ViTs, and hybrids) is the de facto recipe for accuracy, calibration, and robustness, yet two coupled pathologies limit its returns. Prediction-space multicollinearity ill-conditions the meta-learner's Gram matrix, inflating weight variance and producing brittle solutions on a thin manifold. Calibration collapse compounds constituent miscalibration through naive linear stacking, so adding more models can hurt expected calibration error (ECE). Existing remedies, ridge regularization, greedy selection, model soups, and SWAG address at most one of these issues, and none jointly target conditioning and calibration in heterogeneous prediction pools. We introduce CORE-STACK+, a preconditioning pipeline with four components: (i) a kernelized redundancy filter that removes non-linear inter-model dependencies invisible to Pearson correlation, using Centered Kernel Alignment (CKA) [23]; (ii) a <15K-parameter differentiable meta-feature gate that learns per-sample attention over ensemble statistics; (iii) a spectrum-adaptive Ridge penalty lambda^{star}=lmax(Chat)/SNR(Chat) derived from a Marchenko-Pastur signal-noise decomposition, eliminating nested cross-validation; and (iv) a Laplace-approximate Bayesian blender replacing inverse-RMSE heuristics. We prove a PAC-Bayes excess-risk bound that, for the first time, jointly accounts for prediction-space redundancy and meta-learner capacity. Across six benchmarks, CORE-STACK+ delivers +1.8% top-1 on ImageNet-1K, -4.2 mCE on ImageNet-C, +0.9 mIoU on ADE20K, and +1.3 AP on COCO, while reducing retained models by 35-57% and inference FLOPs by up to 41%. ECE improves 2.1times over deep ensembles without post hoc temperature scaling.

Language models are accurate on most long-arithmetic problems, but their accuracy is estimated on samples and carries no guarantee for an unseen input; tool use is exact but moves the computation outside the network. We show that exactness at all input lengths becomes a finite, checkable property of a network when each of its steps reads only a bounded view of an external working memory. In Sketchpad, Qwen3.5-0.8B with a low-rank adapter executes arithmetic procedures on a rendered sheet. Each forward pass reads a three-cell view and emits one step: a digit, a carry or borrow, and the next operation. A fixed runtime moves the view and writes no digit of a result. We prove in Lean that the model is then exact on all inputs of any length if and only if it is exact on the finite set of views the runtime presents, that no finite test suffices when steps also read earlier steps, and that the guarantee tolerates bounded numerical perturbation and extends to compositions of procedures. All 2,446 views pass on the deployment device, a laptop GPU running the model in a web browser. The certified model performs signed addition and subtraction, long multiplication and division, square roots and greatest common divisors at arbitrary length. It solved 200 of 200 pre-registered problems in the browser, and 220 of 220 sums of up to 1,000 digits and products of up to 20 digits, of which GPT-6.1 Sol, the strongest of ten other models, solved 217. A symbolic runtime built on these procedures solves, expands and factors polynomials with every number computed by the model, and a multiplicative gate on the adapter recovers the base model exactly when the sheet is not in use.

We show the formal equivalence of linearised self-attention mechanisms and fast weight controllers from the early '90s, where a ``slow" neural net learns by gradient descent to program the ``fast weights" of another net through sequences of elementary programming instructions which are additive outer products of self-invented activation patterns (today called keys and values). Such Fast Weight Programmers (FWPs) learn to manipulate the contents of a finite memory and dynamically interact with it. We infer a memory capacity limitation of recent linearised softmax attention variants, and replace the purely additive outer products by a delta rule-like programming instruction, such that the FWP can more easily learn to correct the current mapping from keys to values. The FWP also learns to compute dynamically changing learning rates. We also propose a new kernel function to linearise attention which balances simplicity and effectiveness. We conduct experiments on synthetic retrieval problems as well as standard machine translation and language modelling tasks which demonstrate the benefits of our methods.

The widespread adoption of machine learning algorithms necessitates hardware acceleration to ensure efficient performance. This acceleration relies on custom matrix engines that operate on full or reduced-precision floating-point arithmetic. However, conventional floating-point implementations can be power hungry. This paper proposes a method to improve the energy efficiency of the matrix engines used in machine learning algorithm acceleration. Our approach leverages approximate normalization within the floating-point multiply-add units as a means to reduce their hardware complexity, without sacrificing overall machine-learning model accuracy. Hardware synthesis results show that this technique reduces area and power consumption roughly by 16% and 13% on average for Bfloat16 format. Also, the error introduced in transformer model accuracy is 1% on average, for the most efficient configuration of the proposed approach.

In this paper, we present a simple optimization-based preprocessing technique called Weight Magnitude Reduction (MagR) to improve the performance of post-training quantization. For each linear layer, we adjust the pre-trained floating-point weights by solving an ell_infty-regularized optimization problem. This process greatly diminishes the maximum magnitude of the weights and smooths out outliers, while preserving the layer's output. The preprocessed weights are centered more towards zero, which facilitates the subsequent quantization process. To implement MagR, we address the ell_infty-regularization by employing an efficient proximal gradient descent algorithm. Unlike existing preprocessing methods that involve linear transformations and subsequent post-processing steps, which can introduce significant overhead at inference time, MagR functions as a non-linear transformation, eliminating the need for any additional post-processing. This ensures that MagR introduces no overhead whatsoever during inference. Our experiments demonstrate that MagR achieves state-of-the-art performance on the Llama family of models. For example, we achieve a Wikitext2 perplexity of 5.95 on the LLaMA2-70B model for per-channel INT2 weight quantization without incurring any inference overhead.

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

The pre-training cost of large language models (LLMs) is prohibitive. One cutting-edge approach to reduce the cost is zero-shot weight transfer, also known as model growth for some cases, which magically transfers the weights trained in a small model to a large model. However, there are still some theoretical mysteries behind the weight transfer. In this paper, inspired by prior applications of mean field theory to neural network dynamics, we introduce a mean field ansatz to provide a theoretical explanation for weight transfer. Specifically, we propose the row-column (RC) ansatz under the mean field point of view, which describes the measure structure of the weights in the neural network (NN) and admits a close measure dynamic. Thus, the weights of different sizes NN admit a common distribution under proper assumptions, and weight transfer methods can be viewed as sampling methods. We empirically validate the RC ansatz by exploring simple MLP examples and LLMs such as GPT-3 and Llama-3.1. We show the mean-field point of view is adequate under suitable assumptions which can provide theoretical support for zero-shot weight transfer.

Large Language Models (LLMs) suffer from huge number of parameters, which restricts their deployment on edge devices. Weight sharing is one promising solution that encourages weight reuse, effectively reducing memory usage with less performance drop. However, current weight sharing techniques primarily focus on small-scale models like BERT and employ coarse-grained sharing rules, e.g., layer-wise. This becomes limiting given the prevalence of LLMs and sharing an entire layer or block obviously diminishes the flexibility of weight sharing. In this paper, we present a perspective on $textbf{head-wise shareable attention for large language models}. We further propose two memory-efficient methods that share parameters across attention heads, with a specific focus on LLMs. Both of them use the same dynamic strategy to select the shared weight matrices. The first method directly reuses the pre-trained weights without retraining, denoted as DirectShare. The second method first post-trains with constraint on weight matrix similarity and then shares, denoted as PostShare$. Experimental results reveal our head-wise shared models still maintain satisfactory capabilities, demonstrating the feasibility of fine-grained weight sharing applied to LLMs.

Our motivation is to shed light the performance of the widely popular "R-Learner." Like many other methods for estimating conditional average treatment effects (CATEs), R-Learning can be expressed as a weighted pseudo-outcome regression (POR). Previous comparisons of POR techniques have paid careful attention to the choice of pseudo-outcome transformation. However, we argue that the dominant driver of performance is actually the choice of weights. Specifically, we argue that R-Learning implicitly performs an inverse-variance weighted form of POR. These weights stabilize the regression and allow for convenient simplifications of bias terms.

Convolution is a broadly useful operation with applications including signal processing, machine learning, probability, optics, polynomial multiplication, and efficient parsing. Usually, however, this operation is understood and implemented in more specialized forms, hiding commonalities and limiting usefulness. This paper formulates convolution in the common algebraic framework of semirings and semimodules and populates that framework with various representation types. One of those types is the grand abstract template and itself generalizes to the free semimodule monad. Other representations serve varied uses and performance trade-offs, with implementations calculated from simple and regular specifications. Of particular interest is Brzozowski's method for regular expression matching. Uncovering the method's essence frees it from syntactic manipulations, while generalizing from boolean to weighted membership (such as multisets and probability distributions) and from sets to n-ary relations. The classic trie data structure then provides an elegant and efficient alternative to syntax. Pleasantly, polynomial arithmetic requires no additional implementation effort, works correctly with a variety of representations, and handles multivariate polynomials and power series with ease. Image convolution also falls out as a special case.

With the success of deep neural networks, Neural Architecture Search (NAS) as a way of automatic model design has attracted wide attention. As training every child model from scratch is very time-consuming, recent works leverage weight-sharing to speed up the model evaluation procedure. These approaches greatly reduce computation by maintaining a single copy of weights on the super-net and share the weights among every child model. However, weight-sharing has no theoretical guarantee and its impact has not been well studied before. In this paper, we conduct comprehensive experiments to reveal the impact of weight-sharing: (1) The best-performing models from different runs or even from consecutive epochs within the same run have significant variance; (2) Even with high variance, we can extract valuable information from training the super-net with shared weights; (3) The interference between child models is a main factor that induces high variance; (4) Properly reducing the degree of weight sharing could effectively reduce variance and improve performance.

Deep neural network (DNN) deployment has been confined to larger hardware devices due to their expensive computational requirements. This challenge has recently reached another scale with the emergence of large language models (LLMs). In order to reduce both their memory footprint and latency, a promising technique is quantization. It consists in converting floating point representations to low bit-width fixed point representations, usually by assuming a uniform mapping onto a regular grid. This process, referred to in the literature as uniform quantization, may however be ill-suited as most DNN weights and activations follow a bell-shaped distribution. This is even worse on LLMs whose weight distributions are known to exhibit large, high impact, outlier values. In this work, we propose an improvement over the most commonly adopted way to tackle this limitation in deep learning models quantization, namely, non-uniform quantization. NUPES leverages automorphisms to preserve the scalar multiplications. Such transformations are derived from power functions. However, the optimization of the exponent parameter and weight values remains a challenging and novel problem which could not be solved with previous post training optimization techniques which only learn to round up or down weight values in order to preserve the predictive function. We circumvent this limitation with a new paradigm: learning new quantized weights over the entire quantized space. Similarly, we enable the optimization of the power exponent, i.e. the optimization of the quantization operator itself during training by alleviating all the numerical instabilities. The resulting predictive function is compatible with integer-only low-bit inference. We show the ability of the method to achieve state-of-the-art compression rates in both, data-free and data-driven configurations.

We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured weighted sums under exchangeability, extending beyond the classical framework of independent terms. We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability. Along the way, we recover known results in weighted sum of exchangeable random variables and i.i.d. sums of random matrices to the optimal constants. Notably, we develop a sharper concentration bound for combinatorial sum of matrix arrays than the results previously derived from Chatterjee's method of exchangeable pairs. For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models and studying fixed-design sketching methods in federated averaging. We apply our results to these problems, and find that our theoretical predictions are corroborated by numerical evidence.

Large language models have achieved substantial progress in mathematical reasoning, yet their advancement is limited by the scarcity of high-quality, high-difficulty training data. Existing synthesis methods largely rely on transforming human-written templates, limiting both diversity and scalability. We propose MathSmith, a novel framework for synthesizing challenging mathematical problems to enhance LLM reasoning. Rather than modifying existing problems, MathSmith constructs new ones from scratch by randomly sampling concept-explanation pairs from PlanetMath, ensuring data independence and avoiding contamination. To increase difficulty, we design nine predefined strategies as soft constraints during rationales. We further adopts reinforcement learning to jointly optimize structural validity, reasoning complexity, and answer consistency. The length of the reasoning trace generated under autoregressive prompting is used to reflect cognitive complexity, encouraging the creation of more demanding problems aligned with long-chain-of-thought reasoning. Experiments across five benchmarks, categorized as easy & medium (GSM8K, MATH-500) and hard (AIME2024, AIME2025, OlympiadBench), show that MathSmith consistently outperforms existing baselines under both short and long CoT settings. Additionally, a weakness-focused variant generation module enables targeted improvement on specific concepts. Overall, MathSmith exhibits strong scalability, generalization, and transferability, highlighting the promise of high-difficulty synthetic data in advancing LLM reasoning capabilities.

Training compute is increasingly outpacing the availability of high-quality data. This shifts the central challenge from optimal compute allocation to extracting maximum value from limited data. The widely adopted Chinchilla scaling law assumes every training token is unique. This limits its ability to guide pretraining decisions in data-constrained regimes. We model the excess loss under repetition with a simple additive overfitting penalty and find that it accurately describes model behavior. Our scaling law yields qualitatively new compute-optimal allocation advice. Beyond a point, further repetition is counterproductive and compute is better spent on model capacity. We show that following our law's recommended configuration improves performance in data-constrained regimes. Finally, because our one-parameter form isolates overfitting in a single coefficient, it enables direct comparison across training configurations. As a case study, we show that strong weight decay (λ=1.0) reduces this coefficient by approximately 70%, providing a scaling-law explanation for recent findings that optimal weight decay in data-constrained regimes is an order of magnitude larger than standard practice.

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions R^3, position and orientations R^3 {times} S^2, and the group SE(3) itself. Among these, R^3 {times} S^2 is an optimal choice due to the ability to represent directional information, which R^3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

Despite high benchmark scores, Large Language Models (LLMs) often fail simple problem, raising a critical question: Do LLMs learn mathematical principles or merely memorize patterns? Rather than designing increasingly complex benchmarks like recent works, we investigate this using elementary two-integer addition (0 to 2^{64}), probing two core properties: commutativity (A+B=B+A) and compositional generalization (via isomorphic symbolic mappings, e.g., 7 rightarrow y). While state-of-the-art LLMs achieve 73.8-99.8\% accuracy on numerical addition, performance collapses to leq7.5\% under symbolic mapping, indicating failure to generalize learned rules. Non-monotonic performance scaling with digit count and frequent commutativity violations (over 1,700 cases of A+B neq B+A) further support this. Explicitly providing addition rules degrades performance by 81.2\% on average, while self-explanation maintains baseline accuracy, suggesting LLM arithmetic processing is misaligned with human-defined principles. Our findings indicate current LLMs rely on memory pattern over genuine rule learning, highlighting architectural limitations and the need for new approaches to achieve true mathematical reasoning.

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 νapprox 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.

Quantization-Aware Training (QAT) integrates quantization into the training loop, enabling LLMs to learn robust low-bit representations, and is widely recognized as one of the most promising research directions. All current QAT research focuses on minimizing quantization error on full-precision models, where the full-precision accuracy acts as an upper bound (accuracy ceiling). No existing method has even attempted to surpass this ceiling. To break this ceiling, we propose a new paradigm: raising the ceiling (full-precision model), and then still quantizing it efficiently into 2 bits. We propose Fairypm i, the first 2-bit quantization framework for complex-valued LLMs. Specifically, our method leverages the representational advantages of the complex domain to boost full-precision accuracy. We map weights to the fourth roots of unity {pm1, pm i}, forming a perfectly symmetric and information-theoretically optimal 2-bit representation. Importantly, each quantized weight has either a zero real or imaginary part, enabling multiplication-free inference using only additions and element swaps. Experimental results show that Fairypm i outperforms the ceiling of existing 2-bit quantization approaches in terms of both PPL and downstream tasks, while maintaining strict storage and compute efficiency. This work opens a new direction for building highly accurate and practical LLMs under extremely low-bit constraints.

Most Bundle Adjustment (BA) solvers like the Levenberg-Marquardt algorithm require a good initialization. Instead, initialization-free BA remains a largely uncharted territory. The under-explored Variable Projection algorithm (VarPro) exhibits a wide convergence basin even without initialization. Coupled with object space error formulation, recent works have shown its ability to solve small-scale initialization-free bundle adjustment problem. To make such initialization-free BA approaches scalable, we introduce Power Variable Projection (PoVar), extending a recent inverse expansion method based on power series. Importantly, we link the power series expansion to Riemannian manifold optimization. This projective framework is crucial to solve large-scale bundle adjustment problems without initialization. Using the real-world BAL dataset, we experimentally demonstrate that our solver achieves state-of-the-art results in terms of speed and accuracy. To our knowledge, this work is the first to address the scalability of BA without initialization opening new venues for initialization-free structure-from-motion.

Scale has become a main ingredient in obtaining strong machine learning models. As a result, understanding a model's scaling properties is key to effectively designing both the right training setup as well as future generations of architectures. In this work, we argue that scale and training research has been needlessly complex due to reliance on the cosine schedule, which prevents training across different lengths for the same model size. We investigate the training behavior of a direct alternative - constant learning rate and cooldowns - and find that it scales predictably and reliably similar to cosine. Additionally, we show that stochastic weight averaging yields improved performance along the training trajectory, without additional training costs, across different scales. Importantly, with these findings we demonstrate that scaling experiments can be performed with significantly reduced compute and GPU hours by utilizing fewer but reusable training runs.

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

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

Weight oscillation is an undesirable side effect of quantization-aware training, in which quantized weights frequently jump between two quantized levels, resulting in training instability and a sub-optimal final model. We discover that the learnable scaling factor, a widely-used de facto setting in quantization aggravates weight oscillation. In this study, we investigate the connection between the learnable scaling factor and quantized weight oscillation and use ViT as a case driver to illustrate the findings and remedies. In addition, we also found that the interdependence between quantized weights in query and key of a self-attention layer makes ViT vulnerable to oscillation. We, therefore, propose three techniques accordingly: statistical weight quantization (rm StatsQ) to improve quantization robustness compared to the prevalent learnable-scale-based method; confidence-guided annealing (rm CGA) that freezes the weights with high confidence and calms the oscillating weights; and query-key reparameterization (rm QKR) to resolve the query-key intertwined oscillation and mitigate the resulting gradient misestimation. Extensive experiments demonstrate that these proposed techniques successfully abate weight oscillation and consistently achieve substantial accuracy improvement on ImageNet. Specifically, our 2-bit DeiT-T/DeiT-S algorithms outperform the previous state-of-the-art by 9.8% and 7.7%, respectively. Code and models are available at: https://github.com/nbasyl/OFQ.

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.

Intelligent scaling of microservices in cloud platforms is crucial for mitigating escalating compute costs while avoiding service disruptions. Current solutions are limited to the univariate space, typically focusing on CPU usage alone to drive scaling decisions. Moreover, they address the problem as a purely forecasting task, focusing on prediction precision while neglecting the greater risks of underestimation and delays in system responsiveness. Alternative solutions are computationally complex, making them impractical for large-scale, real-time deployments. To address these challenges, we present STARIXNet, a lightweight neural network that guides resource allocation decisions in the multivariate space by capturing spatio-temporal relationships among multiple system metrics. STARIXNet models multiple quasi-dependent attributes, in particular the (S)easonal, (T)emporal, (A)uto-(R)egressive (I)ntegrated, and e(X)ogenous patterns, then implements an aggregation policy to finalize scaling decisions, prioritizing service stability, followed by cost-efficiency, over raw forecast accuracy. We empirically demonstrate the performance of STARIXNet by benchmarking against existing solutions in real-world settings. STARIXNet is deployed for critical production microservices at Walmart achieving tangible savings ranging from 10\% to 50\%, in addition to intangible benefits through improved service stability and customer experience.

A proper initialization of the weights in a neural network is critical to its convergence. Current insights into weight initialization come primarily from linear activation functions. In this paper, I develop a theory for weight initializations with non-linear activations. First, I derive a general weight initialization strategy for any neural network using activation functions differentiable at 0. Next, I derive the weight initialization strategy for the Rectified Linear Unit (RELU), and provide theoretical insights into why the Xavier initialization is a poor choice with RELU activations. My analysis provides a clear demonstration of the role of non-linearities in determining the proper weight initializations.

Continual learning (CL) has traditionally focused on minimizing exemplar memory, a constraint often misaligned with modern systems where GPU time, not storage, is the primary bottleneck. This paper challenges this paradigm by investigating a more realistic regime: one where memory is abundant enough to mitigate forgetting, but full retraining from scratch remains prohibitively expensive. In this practical "middle ground", we find that the core challenge shifts from stability to plasticity, as models become biased toward prior tasks and struggle to learn new ones. Conversely, improved stability allows simple replay baselines to outperform the state-of-the-art methods at a fraction of the GPU cost. To address this newly surfaced trade-off, we propose Weight Space Consolidation, a lightweight method that combines (1) rank-based parameter resets to restore plasticity with (2) weight averaging to enhance stability. Validated on both class-incremental learning with image classifiers and continual instruction tuning with large language models, our approach outperforms strong baselines while matching the low computational cost of replay, offering a scalable alternative to expensive full-retraining. These findings challenge long-standing CL assumptions and establish a new, cost-efficient baseline for real-world CL systems where exemplar memory is no longer the limiting factor.

In this article, we introduce a novel normalization technique for neural network weight matrices, which we term weight conditioning. This approach aims to narrow the gap between the smallest and largest singular values of the weight matrices, resulting in better-conditioned matrices. The inspiration for this technique partially derives from numerical linear algebra, where well-conditioned matrices are known to facilitate stronger convergence results for iterative solvers. We provide a theoretical foundation demonstrating that our normalization technique smoothens the loss landscape, thereby enhancing convergence of stochastic gradient descent algorithms. Empirically, we validate our normalization across various neural network architectures, including Convolutional Neural Networks (CNNs), Vision Transformers (ViT), Neural Radiance Fields (NeRF), and 3D shape modeling. Our findings indicate that our normalization method is not only competitive but also outperforms existing weight normalization techniques from the literature.

Linear time-invariant state space models (SSM) are a classical model from engineering and statistics, that have recently been shown to be very promising in machine learning through the Structured State Space sequence model (S4). A core component of S4 involves initializing the SSM state matrix to a particular matrix called a HiPPO matrix, which was empirically important for S4's ability to handle long sequences. However, the specific matrix that S4 uses was actually derived in previous work for a particular time-varying dynamical system, and the use of this matrix as a time-invariant SSM had no known mathematical interpretation. Consequently, the theoretical mechanism by which S4 models long-range dependencies actually remains unexplained. We derive a more general and intuitive formulation of the HiPPO framework, which provides a simple mathematical interpretation of S4 as a decomposition onto exponentially-warped Legendre polynomials, explaining its ability to capture long dependencies. Our generalization introduces a theoretically rich class of SSMs that also lets us derive more intuitive S4 variants for other bases such as the Fourier basis, and explains other aspects of training S4, such as how to initialize the important timescale parameter. These insights improve S4's performance to 86% on the Long Range Arena benchmark, with 96% on the most difficult Path-X task.

In this paper, we introduce a novel method of neural network weight compression. In our method, we store weight tensors as sparse, quantized matrix factors, whose product is computed on the fly during inference to generate the target model's weights. We use projected gradient descent methods to find quantized and sparse factorization of the weight tensors. We show that this approach can be seen as a unification of weight SVD, vector quantization, and sparse PCA. Combined with end-to-end fine-tuning our method exceeds or is on par with previous state-of-the-art methods in terms of the trade-off between accuracy and model size. Our method is applicable to both moderate compression regimes, unlike vector quantization, and extreme compression regimes.

Decentralized learning often involves a weighted global loss with heterogeneous node weights λ. We revisit two natural strategies for incorporating these weights: (i) embedding them into the local losses to retain a uniform weight (and thus a doubly stochastic matrix), and (ii) keeping the original losses while employing a λ-induced row-stochastic matrix. Although prior work shows that both strategies target the same λ-weighted global loss, it remains unclear whether the Euclidean-space guarantees are tight and what fundamentally differentiates their behaviors. To clarify this, we develop a weighted Hilbert-space framework L^2(λ;R^d) and obtain convergence rates that are strictly tighter than those from standard Euclidean analysis. In this geometry, the row-stochastic matrix becomes self-adjoint whereas the doubly stochastic one does not, creating additional penalty terms that amplify consensus error, thereby slowing convergence. Consequently, the difference in convergence arises not only from spectral gaps but also from these penalty terms. We then derive sufficient conditions under which the row-stochastic design converges faster even with a smaller spectral gap. Finally, by using a Rayleigh-quotient and Loewner-order eigenvalue comparison, we further obtain topology conditions that guarantee this advantage and yield practical topology-design guidelines.

Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces. However, we reveal that existing learnable projection mechanisms cannot ensure stable and balanced assignments from observation points to latent tokens, causing some latent tokens to be over-assigned while others remain underutilized. This limitation further restricts the design of hierarchical architectures, as assignment imbalance is continuously inherited and amplified across latent spaces, eventually causing severe token collapse in deeper spaces. To address these issues, we propose MoNo (Multiscale Optimal Transport Neural Operator), a progressive multiscale neural operator that efficiently solves PDEs on general geometries through stable latent-space construction. At its core is CoTAP (Cross-scale Optimal Transport Assignment and Projection), a novel latent-space construction method that formulates cross-space assignment between adjacent spaces as an entropy-regularized optimal transport problem, thereby constructing balanced bidirectional projections and stable latent spaces. CoTAP also ensures stable information transfer across multiple latent spaces, further enabling multiscale architectures on general geometries, which in turn support more efficient learning of long-range physical interactions. Extensive experiments demonstrate that MoNo outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency. Code is available at https://github.com/ZijiangY1116/MoNo.

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix multiplications in fully-connected layers with Hamilton products of Quaternions, both enjoy parameter savings with only 1/4 learnable parameters and achieve comparable performance in various applications. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions (4D, 8D, and 16D). This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility using arbitrarily 1/n learnable parameters compared with the fully-connected layer counterpart. Experiments of applications to the LSTM and Transformer models on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.

Large language models (LLMs) have demonstrated remarkable capabilities in problem-solving. However, their proficiency in solving mathematical problems remains inadequate. We propose MathScale, a simple and scalable method to create high-quality mathematical reasoning data using frontier LLMs (e.g., {\tt GPT-3.5}). Inspired by the cognitive mechanism in human mathematical learning, it first extracts topics and knowledge points from seed math questions and then build a concept graph, which is subsequently used to generate new math questions. MathScale exhibits effective scalability along the size axis of the math dataset that we generate. As a result, we create a mathematical reasoning dataset (MathScaleQA) containing two million math question-answer pairs. To evaluate mathematical reasoning abilities of LLMs comprehensively, we construct {\sc MwpBench}, a benchmark of Math Word Problems, which is a collection of ten datasets (including GSM8K and MATH) covering K-12, college, and competition level math problems. We apply MathScaleQA to fine-tune open-source LLMs (e.g., LLaMA-2 and Mistral), resulting in significantly improved capabilities in mathematical reasoning. Evaluated on {\sc MwpBench}, MathScale-7B achieves state-of-the-art performance across all datasets, surpassing its best peers of equivalent size by 42.9\% in micro average accuracy and 43.7\% in macro average accuracy, respectively.

Neural networks can approximate complex functions, but they struggle to perform exact arithmetic operations over real numbers. The lack of inductive bias for arithmetic operations leaves neural networks without the underlying logic necessary to extrapolate on tasks such as addition, subtraction, and multiplication. We present two new neural network components: the Neural Addition Unit (NAU), which can learn exact addition and subtraction; and the Neural Multiplication Unit (NMU) that can multiply subsets of a vector. The NMU is, to our knowledge, the first arithmetic neural network component that can learn to multiply elements from a vector, when the hidden size is large. The two new components draw inspiration from a theoretical analysis of recently proposed arithmetic components. We find that careful initialization, restricting parameter space, and regularizing for sparsity is important when optimizing the NAU and NMU. Our proposed units NAU and NMU, compared with previous neural units, converge more consistently, have fewer parameters, learn faster, can converge for larger hidden sizes, obtain sparse and meaningful weights, and can extrapolate to negative and small values.

Large language models (LLMs) have shown remarkable capability in numerous tasks and applications. However, fine-tuning LLMs using high-quality datasets under external supervision remains prohibitively expensive. In response, LLM self-improvement approaches have been vibrantly developed recently. The typical paradigm of LLM self-improvement involves training LLM on self-generated data, part of which may be detrimental and should be filtered out due to the unstable data quality. While current works primarily employs filtering strategies based on answer correctness, in this paper, we demonstrate that filtering out correct but with high distribution shift extent (DSE) samples could also benefit the results of self-improvement. Given that the actual sample distribution is usually inaccessible, we propose a new metric called DS weight to approximate DSE, inspired by the Importance Weighting methods. Consequently, we integrate DS weight with self-consistency to comprehensively filter the self-generated samples and fine-tune the language model. Experiments show that with only a tiny valid set (up to 5\% size of the training set) to compute DS weight, our approach can notably promote the reasoning ability of current LLM self-improvement methods. The resulting performance is on par with methods that rely on external supervision from pre-trained reward models.

This paper discusses the weight parametrization of two standard 1-Lipschitz network architectures, the Almost-Orthogonal-Layers (AOL) and the SDP-based Lipschitz Layers (SLL). It examines their impact on initialization for deep 1-Lipschitz feedforward networks, and discusses underlying issues surrounding this initialization. These networks are mainly used in certifiably robust classification applications to combat adversarial attacks by limiting the impact of perturbations on the classification output. Exact and upper bounds for the parameterized weight variance were calculated assuming a standard Normal distribution initialization; additionally, an upper bound was computed assuming a Generalized Normal Distribution, generalizing the proof for Uniform, Laplace, and Normal distribution weight initializations. It is demonstrated that the weight variance holds no bearing on the output variance distribution and that only the dimension of the weight matrices matters. Additionally, this paper demonstrates that the weight initialization always causes deep 1-Lipschitz networks to decay to zero.

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 .

Sequential learning paradigms pose challenges for gradient-based deep learning due to difficulties incorporating new data and retaining prior knowledge. While Gaussian processes elegantly tackle these problems, they struggle with scalability and handling rich inputs, such as images. To address these issues, we introduce a technique that converts neural networks from weight space to function space, through a dual parameterization. Our parameterization offers: (i) a way to scale function-space methods to large data sets via sparsification, (ii) retention of prior knowledge when access to past data is limited, and (iii) a mechanism to incorporate new data without retraining. Our experiments demonstrate that we can retain knowledge in continual learning and incorporate new data efficiently. We further show its strengths in uncertainty quantification and guiding exploration in model-based RL. Further information and code is available on the project website.

Finding neural network weights that generalize well from small datasets is difficult. A promising approach is to learn a weight initialization such that a small number of weight changes results in low generalization error. We show that this form of meta-learning can be improved by letting the learning algorithm decide which weights to change, i.e., by learning where to learn. We find that patterned sparsity emerges from this process, with the pattern of sparsity varying on a problem-by-problem basis. This selective sparsity results in better generalization and less interference in a range of few-shot and continual learning problems. Moreover, we find that sparse learning also emerges in a more expressive model where learning rates are meta-learned. Our results shed light on an ongoing debate on whether meta-learning can discover adaptable features and suggest that learning by sparse gradient descent is a powerful inductive bias for meta-learning systems.

We introduce PREM (Private Relative Error Multiplicative weight update), a new framework for generating synthetic data that achieves a relative error guarantee for statistical queries under (varepsilon, δ) differential privacy (DP). Namely, for a domain {cal X}, a family {cal F} of queries f : {cal X} to {0, 1}, and ζ> 0, our framework yields a mechanism that on input dataset D cal X^n outputs a synthetic dataset D cal X^n such that all statistical queries in {cal F} on D, namely sum_{x in D} f(x) for f cal F, are within a 1 pm ζ multiplicative factor of the corresponding value on D up to an additive error that is polynomial in log |{cal F}|, log |{cal X}|, log n, log(1/δ), 1/varepsilon, and 1/ζ. In contrast, any (varepsilon, δ)-DP mechanism is known to require worst-case additive error that is polynomial in at least one of n, |{cal F}|, or |{cal X}|. We complement our algorithm with nearly matching lower bounds.

We present any4, a learned 4-bit weight quantization solution for large language models (LLMs) providing arbitrary numeric representations without requiring pre-processing of weights or activations. any4 yields higher accuracy compared to other related 4-bit numeric representation types: int4, fp4 and nf4, as evaluated on a range of model sizes, generations and families (Llama 2, Llama 3, Mistral and Mixtral). While any4 does not require preprocessing of weights or activations, it is also competitive with orthogonal techniques that require such preprocessing (e.g., AWQ and GPTQ). We also experiment with any3 and any2 and show competitiveness at lower bits. Additionally, we show that we can calibrate using a single curated diverse sample rather than hundreds of samples from a dataset as done in most quantization approaches. We also open source tinygemm, a latency optimized GPU matrix multiplication library for LLMs, that implements any4 using a GPU-efficient lookup table strategy along with other common quantization methods. We open source our code at https://github.com/facebookresearch/any4 .

The magnitude of a finite metric space has recently emerged as a novel invariant quantity, allowing to measure the effective size of a metric space. Despite encouraging first results demonstrating the descriptive abilities of the magnitude, such as being able to detect the boundary of a metric space, the potential use cases of magnitude remain under-explored. In this work, we investigate the properties of the magnitude on images, an important data modality in many machine learning applications. By endowing each individual images with its own metric space, we are able to define the concept of magnitude on images and analyse the individual contribution of each pixel with the magnitude vector. In particular, we theoretically show that the previously known properties of boundary detection translate to edge detection abilities in images. Furthermore, we demonstrate practical use cases of magnitude for machine learning applications and propose a novel magnitude model that consists of a computationally efficient magnitude computation and a learnable metric. By doing so, we address the computational hurdle that used to make magnitude impractical for many applications and open the way for the adoption of magnitude in machine learning research.

Given a robust model trained to be resilient to one or multiple types of distribution shifts (e.g., natural image corruptions), how is that "robustness" encoded in the model weights, and how easily can it be disentangled and/or "zero-shot" transferred to some other models? This paper empirically suggests a surprisingly simple answer: linearly - by straightforward model weight arithmetic! We start by drawing several key observations: (1)assuming that we train the same model architecture on both a clean dataset and its corrupted version, resultant weights mostly differ in shallow layers; (2)the weight difference after projection, which we call "Robust Weight Signature" (RWS), appears to be discriminative and indicative of different corruption types; (3)for the same corruption type, the RWSs obtained by one model architecture are highly consistent and transferable across different datasets. We propose a minimalistic model robustness "patching" framework that carries a model trained on clean data together with its pre-extracted RWSs. In this way, injecting certain robustness to the model is reduced to directly adding the corresponding RWS to its weight. We verify our proposed framework to be remarkably (1)lightweight. since RWSs concentrate on the shallowest few layers and we further show they can be painlessly quantized, storing an RWS is up to 13 x more compact than storing the full weight copy; (2)in-situ adjustable. RWSs can be appended as needed and later taken off to restore the intact clean model. We further demonstrate one can linearly re-scale the RWS to control the patched robustness strength; (3)composable. Multiple RWSs can be added simultaneously to patch more comprehensive robustness at once; and (4)transferable. Even when the clean model backbone is continually adapted or updated, RWSs remain as effective patches due to their outstanding cross-dataset transferability.

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