Daily Papers

Nonnegative matrix factorization (NMF) seeks a low-rank approximation X approx UV^T with nonnegative factors and is commonly solved using interior methods that enforce feasibility throughout optimization. We show that such constraint-driven approaches can impede progress in the nonconvex landscape, leading to slow convergence or convergence to suboptimal stationary points. We propose an exterior framework for NMF (eNMF) that separates low-rank approximation from nonnegativity enforcement. Our method initializes from the optimal unconstrained factorization and introduces a rotation procedure that maps unconstrained factors to an exterior point closest to the nonnegative orthant. This viewpoint yields an algorithmic framework in which simple iterative updates converge to KKT-satisfying stationary points on the boundary of the positive orthant. The exterior formulation also enables a geometric interpretation of NMF solutions, clarifying equivalence classes of factorizations under permutation and orthogonal transformations. An intriguing numerical result, involving 400 NMF experiments across both real and synthetic datasets, show that in 99% of the cases, different algorithms tend to converge towards equivalent factor matrices. We benchmark eNMF against 9 state-of-the-art NMF algorithms with 9 initialization schemes across 3 real-world and 2 synthetic datasets. eNMF consistently outperforms all 81 competitors, achieving up to 30% lower reconstruction error under equal-time settings and up to 150% speedup under equal-error settings. The downstream experiments further demonstrate substantial performance gains in audio processing and recommendation tasks, corroborating the practical benefits of the proposed exterior optimization framework. Code is available at https://github.com/roychowdhuryresearch/eNMF

The pursuit of out-of-distribution generalization in Vision-Language-Action (VLA) models is often hindered by catastrophic forgetting of the Vision-Language Model (VLM) backbone during fine-tuning. While co-training with external reasoning data helps, it requires experienced tuning and data-related overhead. Beyond such external dependencies, we identify an intrinsic cause within VLA datasets: modality imbalance, where language diversity is much lower than visual and action diversity. This imbalance biases the model toward visual shortcuts and language forgetting. To address this, we introduce BayesVLA, a Bayesian factorization that decomposes the policy into a visual-action prior, supporting seeing-to-act, and a language-conditioned likelihood, enabling prompt-to-specify. This inherently preserves generalization and promotes instruction following. We further incorporate pre- and post-contact phases to better leverage pre-trained foundation models. Information-theoretic analysis formally validates our effectiveness in mitigating shortcut learning. Extensive experiments show superior generalization to unseen instructions, objects, and environments compared to existing methods. Project page is available at: https://xukechun.github.io/papers/BayesVLA.

Deep learning in de novo protein design has achieved atomic-level fidelity. However, existing models remain largely non-deliberative: they directly synthesize molecular geometries without explicitly reasoning about which residues or interactions are functionally essential. As a result, design decisions are entangled with continuous sampling dynamics, limiting interpretability, controllability, and systematic reuse of biochemical knowledge. We introduce Proteo-R1, a reasoning-guided protein design framework that explicitly decouples molecular understanding from geometric generation. Proteo-R1 adopts a dual-expert architecture in which a multimodal large language model (MLLM) serves as an understanding expert, analyzing protein sequences, structures, and textual context to identify key functional residues that govern binding and specificity. These residue-level decisions are then passed as hard constraints to a separate diffusion-based generation expert, which performs conditional co-design while respecting the fixed interaction anchors. This factorization mirrors how human experts approach molecular engineering: first, reasoning about critical interactions, then optimizing geometry subject to those constraints. By operationalizing reasoning as explicit residue-level commitments rather than latent textual guidance, Proteo-R1 achieves stable, interpretable, and modular integration of LLM reasoning with state-of-the-art geometric generative models. Code, data, and demos are available at https://smiles724.github.io/r1/.

Agents operating in complex software environments benefit from reasoning about the consequences of their actions, as even a single incorrect user interface (UI) operation can derail long, artifact-preserving workflows. This challenge is particularly acute for computer-using scenarios, where real execution does not support counterfactual exploration, making large-scale trial-and-error learning and planning impractical despite the environment being fully digital and deterministic. We introduce the Computer-Using World Model (CUWM), a world model for desktop software that predicts the next UI state given the current state and a candidate action. CUWM adopts a two-stage factorization of UI dynamics: it first predicts a textual description of agent-relevant state changes, and then realizes these changes visually to synthesize the next screenshot. CUWM is trained on offline UI transitions collected from agents interacting with real Microsoft Office applications, and further refined with a lightweight reinforcement learning stage that aligns textual transition predictions with the structural requirements of computer-using environments. We evaluate CUWM via test-time action search, where a frozen agent uses the world model to simulate and compare candidate actions before execution. Across a range of Office tasks, world-model-guided test-time scaling improves decision quality and execution robustness.

Out-of-town recommendation is designed for those users who leave their home-town areas and visit the areas they have never been to before. It is challenging to recommend Point-of-Interests (POIs) for out-of-town users since the out-of-town check-in behavior is determined by not only the user's home-town preference but also the user's travel intention. Besides, the user's travel intentions are complex and dynamic, which leads to big difficulties in understanding such intentions precisely. In this paper, we propose a TRAvel-INtention-aware Out-of-town Recommendation framework, named TRAINOR. The proposed TRAINOR framework distinguishes itself from existing out-of-town recommenders in three aspects. First, graph neural networks are explored to represent users' home-town check-in preference and geographical constraints in out-of-town check-in behaviors. Second, a user-specific travel intention is formulated as an aggregation combining home-town preference and generic travel intention together, where the generic travel intention is regarded as a mixture of inherent intentions that can be learned by Neural Topic Model (NTM). Third, a non-linear mapping function, as well as a matrix factorization method, are employed to transfer users' home-town preference and estimate out-of-town POI's representation, respectively. Extensive experiments on real-world data sets validate the effectiveness of the TRAINOR framework. Moreover, the learned travel intention can deliver meaningful explanations for understanding a user's travel purposes.

Large language models (LLMs) face low hardware efficiency during decoding, especially for long-context reasoning tasks. This paper introduces Step-3, a 321B-parameter VLM with hardware-aware model-system co-design optimized for minimizing decoding costs. Step-3 innovates in two key dimensions: (1) A novel Multi-Matrix Factorization Attention (MFA) mechanism that significantly reduces both KV cache size and computation while maintaining high attention expressiveness, and (2) Attention-FFN Disaggregation (AFD), a distributed inference system that decouples attention and Feed-Forward Network (FFN) layers into specialized subsystems. This co-design achieves unprecedented cost efficiency: Step-3 significantly reduces theoretical decoding costs compared with models like DeepSeek-V3 and Qwen3 MoE 235B, with the gains widening at longer context. Step-3 achieves low cost while activating 38B parameters per token (more than DeepSeek-V3 and Qwen3 MoE 235B), demonstrating that hardware-aligned attention arithmetic intensity, MoE sparsity, and AFD are critical to cost-effectiveness. We perform a head-to-head comparison with DeepSeek-V3 in its favorable scenarios. Our implementation on Hopper GPUs achieves a decoding throughput of up to 4,039 tokens per second per GPU under 50ms TPOT SLA (4K context, FP8, no MTP). It is higher than DeepSeek-V3's 2,324 in the same setup and sets a new Pareto frontier for LLM decoding.

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

We introduce Seed-TTS, a family of large-scale autoregressive text-to-speech (TTS) models capable of generating speech that is virtually indistinguishable from human speech. Seed-TTS serves as a foundation model for speech generation and excels in speech in-context learning, achieving performance in speaker similarity and naturalness that matches ground truth human speech in both objective and subjective evaluations. With fine-tuning, we achieve even higher subjective scores across these metrics. Seed-TTS offers superior controllability over various speech attributes such as emotion and is capable of generating highly expressive and diverse speech for speakers in the wild. Furthermore, we propose a self-distillation method for speech factorization, as well as a reinforcement learning approach to enhance model robustness, speaker similarity, and controllability. We additionally present a non-autoregressive (NAR) variant of the Seed-TTS model, named Seed-TTS_DiT, which utilizes a fully diffusion-based architecture. Unlike previous NAR-based TTS systems, Seed-TTS_DiT does not depend on pre-estimated phoneme durations and performs speech generation through end-to-end processing. We demonstrate that this variant achieves comparable performance to the language model-based variant and showcase its effectiveness in speech editing. We encourage readers to listen to demos at https://bytedancespeech.github.io/seedtts_tech_report.

Code based Language Models (LMs) have shown very promising results in the field of software engineering with applications such as code refinement, code completion and generation. However, the task of time and space complexity classification from code has not been extensively explored due to a lack of datasets, with prior endeavors being limited to Java. In this project, we aim to address these gaps by creating a labelled dataset of code snippets spanning multiple languages (Python and C++ datasets currently, with C, C#, and JavaScript datasets being released shortly). We find that existing time complexity calculation libraries and tools only apply to a limited number of use-cases. The lack of a well-defined rule based system motivates the application of several recently proposed code-based LMs. We demonstrate the effectiveness of dead code elimination and increasing the maximum sequence length of LMs. In addition to time complexity, we propose to use LMs to find space complexities from code, and to the best of our knowledge, this is the first attempt to do so. Furthermore, we introduce a novel code comprehension task, called cross-language transfer, where we fine-tune the LM on one language and run inference on another. Finally, we visualize the activation of the attention fed classification head of our LMs using Non-negative Matrix Factorization (NMF) to interpret our results.

In recent years, the scaling laws of recommendation models have attracted increasing attention, which govern the relationship between performance and parameters/FLOPs of recommenders. Currently, there are three mainstream architectures for achieving scaling in recommendation models, namely attention-based, TokenMixer-based, and factorization-machine-based methods, which exhibit fundamental differences in both design philosophy and architectural structure. In this paper, we propose a unified scaling architecture for recommendation systems, namely UniMixer, to improve scaling efficiency and establish a unified theoretical framework that unifies the mainstream scaling blocks. By transforming the rule-based TokenMixer to an equivalent parameterized structure, we construct a generalized parameterized feature mixing module that allows the token mixing patterns to be optimized and learned during model training. Meanwhile, the generalized parameterized token mixing removes the constraint in TokenMixer that requires the number of heads to be equal to the number of tokens. Furthermore, we establish a unified scaling module design framework for recommender systems, which bridges the connections among attention-based, TokenMixer-based, and factorization-machine-based methods. To further boost scaling ROI, a lightweight UniMixing module is designed, UniMixing-Lite, which further compresses the model parameters and computational cost while significantly improve the model performance. The scaling curves are shown in the following figure. Extensive offline and online experiments are conducted to verify the superior scaling abilities of UniMixer.

Current instruction-guided video editing models struggle to simultaneously balance precise semantic modifications with faithful motion preservation. While existing approaches rely on injecting explicit external priors (e.g., VLM features or structural conditions) to mitigate these issues, this reliance severely bottlenecks model robustness and generalization. To overcome this limitation, we present SAMA (factorized Semantic Anchoring and Motion Alignment), a framework that factorizes video editing into semantic anchoring and motion modeling. First, we introduce Semantic Anchoring, which establishes a reliable visual anchor by jointly predicting semantic tokens and video latents at sparse anchor frames, enabling purely instruction-aware structural planning. Second, Motion Alignment pre-trains the same backbone on motion-centric video restoration pretext tasks (cube inpainting, speed perturbation, and tube shuffle), enabling the model to internalize temporal dynamics directly from raw videos. SAMA is optimized with a two-stage pipeline: a factorized pre-training stage that learns inherent semantic-motion representations without paired video-instruction editing data, followed by supervised fine-tuning on paired editing data. Remarkably, the factorized pre-training alone already yields strong zero-shot video editing ability, validating the proposed factorization. SAMA achieves state-of-the-art performance among open-source models and is competitive with leading commercial systems (e.g., Kling-Omni). Code, models, and datasets will be released.

Production machine learning (ML) stacks often split graph compilation and kernel execution across different layers and languages, making backend behavior, deployment guarantees, and performance fallbacks hard to reason about end-to-end. RLX addresses this gap with a single Rust codebase that combines compiler and runtime roles around one primitive-level, three-level intermediate representation (IR), plus a transparent dispatch contract that resolves each operator to native, common-IR, or rewritten lowering and fails compilation when legalization is not possible. The same IR targets fourteen runtime devices (cpu, metal, mlx, ane, cuda, rocm, oneapi, tpu, hexagon, gpu, vulkan, opengl, directx, webgpu) and two specialty codegen paths (Cortex-M INT8 and FPGA), ingests safetensors, GGUF, ONNX, and rten formats, supports F16/BF16/F64/C64 and quantized INT4/INT8 flows with AMP/PTQ/QAT, and scales via tensor-/pipeline-parallel collectives over TCP and RDMA transports. Beyond neural workloads, RLX also extends to scientific/physics-style domains through sparse and dense linear algebra extensions (e.g., CSR LU/CG/matvec and LAPACK- backed factorizations) and 3D Gaussian splatting operators. We evaluate RLX against PyTorch, TensorFlow, JAX, candle, burn, tch, rten, MLX, CoreML, IREE, Glow, TensorRT, and tinygrad under identical input generation and p50 measurement methodology on one host. On all-MiniLM-L6-v2, RLX-Metal is fastest at every batch (e.g., 16.6 ms at batch 32 vs. PyTorch-MPS 26.7 ms). In the MNIST training table, RLX also has the top-throughput entry (graph-fused MLP: 946,487 img/s), above NumPy+BLAS (787,349 img/s), while retaining 100% top-1 parity on reference checks (e.g., Qwen3).

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.

We introduce ExTernD (Expanded-rank Ternary Decomposition), a post-training factorization of each LLM weight matrix A in R^{m times n} into A approx B diag(D) C with ternary factors B in {-1,0,+1}^{m times k}, C in {-1,0,+1}^{k times n} and a real scale vector D in R^k. The inner rank k = μmin(m,n) is deliberately expanded beyond full rank (μ> 1), so that components past full rank correct the quantization error of earlier ones. We prove the residual decreases monotonically in k and can be driven below any varepsilon > 0: ExTernD approaches bf16 accuracy arbitrarily closely, which no ternary scheme with a fixed plane count can do. Memory and compute scale continuously with μ, and factor sparsity continuously with a threshold τ, so an accuracy target is hit exactly rather than rounded to the next bit-width. ExTernD matches Q4_K's per-matrix accuracy at 5.2-5.5 effective bpw (5.1-5.5 with importance weighting) on Gemma-4-E2B and Qwen3.5-4B, and a full Qwen3.5-4B conversion at μ= 3 reaches 10.10 wikitext-2 perplexity against 9.78 for bf16 (+3.2%), placing it near the Q4_K/Q5_K accuracy band at ~5.7 effective bpw.

As large language model (LLM) inference demands ever-greater resources, there is a rapid growing trend of using low-bit weights to shrink memory usage and boost inference efficiency. However, these low-bit LLMs introduce the need for mixed-precision matrix multiplication (mpGEMM), which is a crucial yet under-explored operation that involves multiplying lower-precision weights with higher-precision activations. Unfortunately, current hardware does not natively support mpGEMM, resulting in indirect and inefficient dequantization-based implementations. To address the mpGEMM requirements in low-bit LLMs, we explored the lookup table (LUT)-based approach for mpGEMM. However, a conventional LUT implementation falls short of its potential. To fully harness the power of LUT-based mpGEMM, we introduce LUT Tensor Core, a software-hardware co-design optimized for low-bit LLM inference. Specifically, we introduce software-based operator fusion and table symmetrization techniques to optimize table precompute and table storage, respectively. Then, LUT Tensor Core proposes the hardware design featuring an elongated tiling shape design to enhance table reuse and a bit-serial design to support various precision combinations in mpGEMM. Moreover, we design an end-to-end compilation stack with new instructions for LUT-based mpGEMM, enabling efficient LLM compilation and optimizations. The evaluation on low-bit LLMs (e.g., BitNet, LLAMA) shows that LUT Tensor Core achieves more than a magnitude of improvements on both compute density and energy efficiency.

Large language models (LLMs) are increasingly deployed on edge devices. To meet strict resource constraints, real-world deployment has pushed LLM quantization from 8-bit to 4-bit, 2-bit, and now 1.58-bit. Combined with lookup table (LUT)-based inference, CPUs run these ultra-low-bit LLMs even faster than NPUs, opening new opportunities for ubiquitous on-device intelligence. However, this paper identifies that LUT-based inference underutilizes memory bandwidth during parallel inference, which is required for prefilling, test-time scaling, and other multi-token scenarios. The root cause is the scalar LUT paradigm, which performs repetitive and non-contiguous memory accesses for each token. To solve the issue, we propose vector LUT, a new lookup paradigm that constructs a unified LUT across parallel tokens, and performs a single 1 rightarrow N lookup per index. To realize it efficiently, we further introduce (1) Vector LUT-Centric Tensor Layout, and (2) Cache-Aware Streamed Lookup techniques. Evaluations on 5 edge devices across 3 LLMs show that Vec-LUT outperforms state-of-the-art baselines by up to 4.2times. Our implementation is integrated into llama.cpp. The code is available at https://github.com/Cipherxzc/vlut.cpp.

Recent research has demonstrated that Feed-Forward Networks (FFNs) in Large Language Models (LLMs) play a pivotal role in storing diverse linguistic and factual knowledge. Conventional methods frequently face challenges due to knowledge confusion stemming from their monolithic and redundant architectures, which calls for more efficient solutions with minimal computational overhead, particularly for LLMs. In this paper, we explore the FFN computation paradigm in LLMs and introduce FactorLLM, a novel approach that decomposes well-trained dense FFNs into sparse sub-networks without requiring any further modifications, while maintaining the same level of performance. Furthermore, we embed a router from the Mixture-of-Experts (MoE), combined with our devised Prior-Approximate (PA) loss term that facilitates the dynamic activation of experts and knowledge adaptation, thereby accelerating computational processes and enhancing performance using minimal training data and fine-tuning steps. FactorLLM thus enables efficient knowledge factorization and activates select groups of experts specifically tailored to designated tasks, emulating the interactive functional segmentation of the human brain. Extensive experiments across various benchmarks demonstrate the effectiveness of our proposed FactorLLM which achieves comparable performance to the source model securing up to 85% model performance while obtaining over a 30% increase in inference speed. Code: https://github.com/zhenwuweihe/FactorLLM.

Scaling autoregressive large language models (LLMs) has driven unprecedented progress but comes with vast computational costs. In this work, we tackle these costs by leveraging unstructured sparsity within an LLM's feedforward layers, the components accounting for most of the model parameters and execution FLOPs. To achieve this, we introduce a new sparse packing format and a set of CUDA kernels designed to seamlessly integrate with the optimized execution pipelines of modern GPUs, enabling efficient sparse computation during LLM inference and training. To substantiate our gains, we provide a quantitative study of LLM sparsity, demonstrating that simple L1 regularization can induce over 99% sparsity with negligible impact on downstream performance. When paired with our kernels, we show that these sparsity levels translate into substantial throughput, energy efficiency, and memory usage benefits that increase with model scale. We will release all code and kernels under an open-source license to promote adoption and accelerate research toward establishing sparsity as a practical axis for improving the efficiency and scalability of modern foundation models.

This paper explores the relationship between matrix factorizations and linear matrix equations. It shows that every matrix factorization defines two hidden projectors, one for the column space and one for the row space of a matrix, and how to calculate them. The projectors can be applied to solve linear matrix equations, generate low-rank approximations, or design randomized matrix algorithms. But also, as demonstrated, they can be applied in cryptography to encrypt and decrypt messages. The paper discusses some of the security implications of this application and leaves some questions open for further investigation. The basic concepts are illustrated with source code listings. Finally, this work shares some personal reflections on the meaning and importance of understanding in the time of the artificial intelligence revolution.

Modern transformers pair impressive capabilities with substantial memory and compute demands. Low-rank weight factorization reduces both while keeping the matrices dense, and thus efficient on standard hardware. Existing methods, however, choose the subspace to remove from each weight matrix with local closed-form criteria: activation energy, layer-wise reconstruction error, or a quadratic approximation of the loss. These criteria ignore how errors propagate through the network, so at high compression the errors compound with depth and performance collapses. We introduce Learnable Subspace Projections (LSP), which instead learns the subspaces to discard end-to-end. Each linear layer, or tied group of layers that read the same activations, is assigned an orthogonal projector. All projectors are optimized jointly against a global objective--the KL divergence to the dense model's output distribution or the model's original training loss--while the pretrained weights remain frozen. Projectors are initialized from a whitened SVD truncation, and ranks are allocated by the output KL each projector induces per parameter saved. After training, the projectors merge into standard low-rank factors, with each tied group sharing one factor. In attention, this also lets the model cache one narrow latent in place of full keys and values. Across LLMs (OPT-125M/1.3B, Qwen3-4B, Llama-2-7B) and ViT-B/16, LSP outperforms baselines, and its advantage widens as compression increases. At -70% compression, LSP brings Llama-2-7B to 10.9 WikiText-2 perplexity and 42.2% mean zero-shot accuracy, versus 13.3 and 36.0% for the strongest baseline. The factorized model decodes up to 1.6x faster than the dense model at small batch sizes, and aching the shared latent shrinks the combined memory of weights and KV cache by 13.5x at a 128k-token context, versus at most 6.5x for untied baseline factorizations.

Weight-Decomposed Low-Rank Adaptation (DoRA) extends LoRA by decoupling weight magnitude from direction, but its forward pass requires the row-wise norm of W + sBA, a computation that every major framework we surveyed implements by materializing the dense [d_out, d_in] product BA. At d_in = 8192 and rank r = 384, a single module's norm requires about 512 MB of transient working memory in bf16, making high-rank DoRA costly and often infeasible on common single-GPU setups once hundreds of adapted modules and checkpointing are involved. We present two systems contributions. A factored norm decomposes the squared norm into base, cross, and Gram terms computable through O(d_out r + r^2) intermediates, eliminating the dense product. Fused Triton kernels collapse the four-kernel DoRA composition into a single pass, reducing memory traffic by about 4x and using a numerically stable form that avoids catastrophic cancellation in the near-unity rescaling regime where magnitude scales concentrate in practice. Across six 8-32B vision-language models (VLMs) on three NVIDIA GPUs (RTX 6000 PRO, H200, B200) at r = 384 in bf16, the fused implementation is 1.5-2.0x faster than Hugging Face PEFT's DoRA implementation for inference and 1.5-1.9x faster for gradient computation (optimizer step excluded), with up to 7 GB lower peak VRAM. Microbenchmarks on six GPUs spanning four architecture generations (L40S, A100, RTX 6000 PRO, H200, B200, B300) confirm 1.5-2.7x compose-kernel speedup. Final-logit cosine similarity exceeds 0.9999 across all model/GPU pairs, and multi-seed training curves match within 7.1 x 10^-4 mean per-step loss delta over 2000 steps.

Recent advances in self-supervised learning and the Transformer architecture have significantly improved natural language processing (NLP), achieving remarkably low perplexity. However, the growing size of NLP models introduces a memory wall problem during the generation phase. To mitigate this issue, recent efforts have focused on quantizing model weights to sub-4-bit precision while preserving full precision for activations, resulting in practical speed-ups during inference on a single GPU. However, these improvements primarily stem from reduced memory movement, which necessitates a resource-intensive dequantization process rather than actual computational reduction. In this paper, we introduce LUT-GEMM, an efficient kernel for quantized matrix multiplication, which not only eliminates the resource-intensive dequantization process but also reduces computational costs compared to previous kernels for weight-only quantization. Furthermore, we proposed group-wise quantization to offer a flexible trade-off between compression ratio and accuracy. The impact of LUT-GEMM is facilitated by implementing high compression ratios through low-bit quantization and efficient LUT-based operations. We show experimentally that when applied to the OPT-175B model with 3-bit quantization, LUT-GEMM substantially accelerates token generation latency, achieving a remarkable 2.1times improvement on a single GPU when compared to OPTQ, which relies on the costly dequantization process.

Activation sparsity denotes the existence of substantial weakly-contributed elements within activation outputs that can be eliminated, benefiting many important applications concerned with large language models (LLMs). Although promoting greater activation sparsity within LLMs deserves deep studies, existing works lack comprehensive and quantitative research on the correlation between activation sparsity and potentially influential factors. In this paper, we present a comprehensive study on the quantitative scaling properties and influential factors of the activation sparsity within decoder-only Transformer-based LLMs. Specifically, we propose PPL-p% sparsity, a precise and performance-aware activation sparsity metric that is applicable to any activation function. Through extensive experiments, we find several important phenomena. Firstly, different activation functions exhibit comparable performance but opposite training-time sparsity trends. The activation ratio (i.e., 1-sparsity ratio) evolves as a convergent increasing power-law and decreasing logspace power-law with the amount of training data for SiLU-activated and ReLU-activated LLMs, respectively. These demonstrate that ReLU is more efficient as the activation function than SiLU and can leverage more training data to improve activation sparsity. Secondly, the activation ratio linearly increases with the width-depth ratio below a certain bottleneck point, indicating the potential advantage of a deeper architecture at a fixed parameter scale. Finally, at similar width-depth ratios, we surprisingly find that the limit value of activation sparsity varies weakly with the parameter scale, i.e., the activation patterns within LLMs are insensitive to the parameter scale. These empirical laws towards LLMs with greater activation sparsity have important implications for making LLMs more efficient and interpretable.

Recent advancements in autoregressive Large Language Models (LLMs) have achieved significant milestones, largely attributed to their scalability, often referred to as the "scaling law". Inspired by these achievements, there has been a growing interest in adapting LLMs for Recommendation Systems (RecSys) by reformulating RecSys tasks into generative problems. However, these End-to-End Generative Recommendation (E2E-GR) methods tend to prioritize idealized goals, often at the expense of the practical advantages offered by traditional Deep Learning based Recommendation Models (DLRMs) in terms of in features, architecture, and practices. This disparity between idealized goals and practical needs introduces several challenges and limitations, locking the scaling law in industrial RecSys. In this paper, we introduce a large user model (LUM) that addresses these limitations through a three-step paradigm, designed to meet the stringent requirements of industrial settings while unlocking the potential for scalable recommendations. Our extensive experimental evaluations demonstrate that LUM outperforms both state-of-the-art DLRMs and E2E-GR approaches. Notably, LUM exhibits excellent scalability, with performance improvements observed as the model scales up to 7 billion parameters. Additionally, we have successfully deployed LUM in an industrial application, where it achieved significant gains in an A/B test, further validating its effectiveness and practicality.

The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon's efficiency relies on fast, approximate orthogonalization, yet all prior theoretical work analyzes an idealized, computationally intractable version assuming exact SVD-based updates. This work moves beyond the ideal by providing the first analysis of the inexact orthogonalized update at Muon's core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure (e.g., the number of Newton-Schulz steps) from an implementation detail to a critical parameter that must be co-tuned with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes.

This monograph develops a unified account of training and inference in Power Law Decoder Representation language models (PLDR-LLMs). Exact finite work identities decompose changes in the absolute energy of the row-centered learned map into parameter contributions, signed interactions, and numerical observation defects. Positive affine blocking retains restarts at the row-constant face, while the augmented AdamW state supplies the complete dynamical description. Predictive renormalization acts on the complete conditional training law for a single pass over distinct corpus target blocks, retaining optimizer memory, remaining data, schedule, and numerical policy. Autonomous reductions require closure; approximate reductions carry successor and emission errors. Finite-population covariance, matched physical clocks, matrix fluxes, and signed temporal energy connect row dynamics to model-wide observations. Absolute row collapse, relative row concentration, operator stabilization, and predictive accuracy are distinguished. Experiments reveal observer and optimizer dependence, reject the tested autonomous row-state candidates, and support finite conditional prediction and state-specific operator reduction. Independent single-pass families exhibit moving finite fluctuation regions without establishing a thermodynamic critical class. Conditional symmetry, head limits, covariance flows, and readout error budgets specify assumptions needed to transfer scaling laws to inference. The theory separates exact identities, conditional dynamical claims, and finite empirical findings, with proofs, selected formal checks, and compact numerical evidence.

Layer factorization has emerged as a widely used technique for training memory-efficient neural networks. However, layer factorization methods face several challenges, particularly a lack of robustness during the training process. To overcome this limitation, dynamical low-rank training methods have been developed, utilizing robust time integration techniques for low-rank matrix differential equations. Although these approaches facilitate efficient training, they still depend on computationally intensive QR and singular value decompositions of matrices with small rank. In this work, we introduce a novel low-rank training method that reduces the number of required QR decompositions. Our approach integrates an augmentation step into a projector-splitting scheme, ensuring convergence to a locally optimal solution. We provide a rigorous theoretical analysis of the proposed method and demonstrate its effectiveness across multiple benchmarks.

Large language models (LLMs) have been widely applied but face challenges in efficient inference. While quantization methods reduce computational demands, ultra-low bit quantization with arbitrary precision is hindered by limited GPU Tensor Core support and inefficient memory management, leading to suboptimal acceleration. To address these challenges, we propose a comprehensive acceleration scheme for arbitrary precision LLMs. At its core, we introduce a novel bipolar-INT data format that facilitates parallel computing and supports symmetric quantization, effectively reducing data redundancy. Building on this, we implement an arbitrary precision matrix multiplication scheme that decomposes and recovers matrices at the bit level, enabling flexible precision while maximizing GPU Tensor Core utilization. Furthermore, we develop an efficient matrix preprocessing method that optimizes data layout for subsequent computations. Finally, we design a data recovery-oriented memory management system that strategically utilizes fast shared memory, significantly enhancing kernel execution speed and minimizing memory access latency. Experimental results demonstrate our approach's effectiveness, with up to 2.4\times speedup in matrix multiplication compared to NVIDIA's CUTLASS. When integrated into LLMs, we achieve up to 6.7\times inference acceleration. These improvements significantly enhance LLM inference efficiency, enabling broader and more responsive applications of LLMs.

Randomized linear algebra (RLA) algorithms are a modern class of numerical linear algebra techniques that play an essential role in scientific computing and machine learning, with broad and growing adoption. However, their discovery remains mostly a manual process that requires deep expert knowledge and inspiration. While Reinforcement Learning (RL) offers a pathway to automation, standard approaches struggle with sparse reward landscapes and vast search spaces inherent to high-performing RLA algorithms. In this paper, we present RL4RLA, a general RL framework that automates the discovery of interpretable, symbolic RLA algorithms. Unlike black-box approaches, our method builds explicit algorithms from basic linear algebra primitives, ensuring verifiable and implementable representations. To enable efficient discovery, we introduce: (1) a numerical curriculum that progressively increments problem difficulty to encode inductive bias specific to the RLA domain; (2) Monte Carlo Graph Search, which optimizes exploration by identifying and merging equivalent partial algorithms. We demonstrate that RL4RLA rediscovers state-of-the-art methods, including sketch-and-precondition solvers, Randomized Kaczmarz, and Newton Sketch, and can be targeted to produce algorithms optimized for specific trade-offs between accuracy, speed, and stability. Code is available at https://github.com/Tim-Xiong/RL4RLA.

Retrieval-augmented language models such as Fusion-in-Decoder are powerful, setting the state of the art on a variety of knowledge-intensive tasks. However, they are also expensive, due to the need to encode a large number of retrieved passages. Some work avoids this cost by pre-encoding a text corpus into a memory and retrieving dense representations directly. However, pre-encoding memory incurs a severe quality penalty as the memory representations are not conditioned on the current input. We propose LUMEN, a hybrid between these two extremes, pre-computing the majority of the retrieval representation and completing the encoding on the fly using a live encoder that is conditioned on the question and fine-tuned for the task. We show that LUMEN significantly outperforms pure memory on multiple question-answering tasks while being much cheaper than FiD, and outperforms both for any given compute budget. Moreover, the advantage of LUMEN over FiD increases with model size.

Exploiting activation sparsity is a promising approach to significantly accelerating the inference process of large language models (LLMs) without compromising performance. However, activation sparsity is determined by activation functions, and commonly used ones like SwiGLU and GeGLU exhibit limited sparsity. Simply replacing these functions with ReLU fails to achieve sufficient sparsity. Moreover, inadequate training data can further increase the risk of performance degradation. To address these challenges, we propose a novel dReLU function, which is designed to improve LLM activation sparsity, along with a high-quality training data mixture ratio to facilitate effective sparsification. Additionally, we leverage sparse activation patterns within the Feed-Forward Network (FFN) experts of Mixture-of-Experts (MoE) models to further boost efficiency. By applying our neuron sparsification method to the Mistral and Mixtral models, only 2.5 billion and 4.3 billion parameters are activated per inference iteration, respectively, while achieving even more powerful model performance. Evaluation results demonstrate that this sparsity achieves a 2-5x decoding speedup. Remarkably, on mobile phones, our TurboSparse-Mixtral-47B achieves an inference speed of 11 tokens per second. Our models are available at https://huggingface.co/PowerInfer

The μ-parameterization (μP) provides a principled foundation for large language model (LLM) training by prescribing width-independent learning dynamics, which in turn enables predictable scaling behavior and robust hyperparameter transfer across model sizes. A central requirement of μP is the satisfaction of certain spectral conditions on weight matrices, which ensure consistent feature learning and optimization behavior as model width grows. While these conditions are well understood in theory, guaranteeing their validity in practical training for matrix-based optimizers such as Muon is still under studied. Existing works that study Muon under μP exhibit important limitations: they either do not ensure that the spectral conditions hold throughout the entire training horizon, or require repeated spectral normalization (or Newton-Schulz iterations) applied to both weights and updates, leading to significant computational overhead and reduced practicality. In this work, we show how to reliably guarantee the spectral conditions required by μP for Muon during the entire training process. Our key insight is that for moderately large models, maintaining spectral control at the level of optimizer updates alone is sufficient to preserve μP-compatible scaling, eliminating the need for explicit spectral normalization of the weights. Based on this principle, we develop a variant of Muon, namely Muon++, that satisfies spectral condition throughout the training process. Our results bridge the gap between the theoretical promises of μP and the practical deployment of matrix-based optimizers in long-horizon training. We also take the first step towards an adaptive spectral condition by incorporating data-dependent effects, making it better suited for long-horizon LLM training.

This paper makes a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which results in the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. A second important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

End-to-end document parsers provide a unified interface, but serialize page layouts and regional contents into one autoregressive sequence. This formulation forces independent regions onto a decoding path whose length grows with the total content, whereas crop-based two-stage parsers expose region-level parallelism at the cost of repeated visual prefills and fragmented page context. To retain full-page context while removing dependencies, we propose PaDoc, a layout-grounded parser that treats the predicted layout as a branching structure over a shared page representation. Under a region-sufficiency assumption, we derive a prefix-conditioned factorization in which the layout stream and regional content branches advance concurrently, reducing the decoding depth to the longest layout-content path. We realize this factorization within a single MLLM: packed variable-length ancestor attention preserves the visibility under standard next-token training, while masked parallel decoding creates branches that the evaluated vLLM backend serves as concurrent requests with cache-resident shared-prefix reuse. On OmniDocBench Full, PaDoc attains an Overall layout F1 of 91.1 and, among end-to-end parsers, a top-tier Overall score of 94.24 together with the best Text Edit (0.038) and Formula CDM (95.59). On a 384-page subset and one A800 GPU, it is the fastest end-to-end parser at five concurrency levels, improving valid-page throughput by 67.4-118% and reducing P95 latency by 39.2-54.9% relative to a same-backbone Sequential SFT baseline. Code is available at https://github.com/Longin-Yu/Padoc

Low-rank factorization is widely used to compress neural networks, but modern models are often not naturally amenable to aggressive factorization without significant accuracy loss. Existing training-time low-rank regularizers can improve compressibility, but they often require SVDs of large weight matrices, modify the model architecture (introducing additional trainable parameters), or rely on stateful cached quantities. To address these limitations, we introduce SLORR, a simple, stateless, and architecture-preserving framework for in-training low-rank regularization, instantiated with two main variants based on the Hoyer sparsity metric and the nuclear norm. SLORR directly regularizes the original weight matrices using GPU-friendly approximations for the forward and backward passes of the regularizers, for which we provide approximation guarantees. We first evaluate SLORR on ImageNet-1K across short-horizon continued training of ResNet-50, ViT-B/16, and ViT-L/16, and pretraining of ResNet-18, where SLORR induces compressibility while introducing less than 8% training overhead. We further evaluate SLORR-Hoyer in LLM pretraining at 135M and 560M scales: SLORR-trained compressed models preserve performance substantially better than unregularized models while adding less than 1% average training overhead.

Most conventional recommendation methods (e.g., matrix factorization) represent user profiles as high-dimensional vectors. Unfortunately, these vectors lack interpretability and steerability, and often perform poorly in cold-start settings. To address these shortcomings, we explore the use of user profiles that are represented as human-readable text. We propose the Language-based Factorization Model (LFM), which is essentially an encoder/decoder model where both the encoder and the decoder are large language models (LLMs). The encoder LLM generates a compact natural-language profile of the user's interests from the user's rating history. The decoder LLM uses this summary profile to complete predictive downstream tasks. We evaluate our LFM approach on the MovieLens dataset, comparing it against matrix factorization and an LLM model that directly predicts from the user's rating history. In cold-start settings, we find that our method can have higher accuracy than matrix factorization. Furthermore, we find that generating a compact and human-readable summary often performs comparably with or better than direct LLM prediction, while enjoying better interpretability and shorter model input length. Our results motivate a number of future research directions and potential improvements.

Despite their increasing popularity and success in a variety of supervised learning problems, deep neural networks are extremely hard to interpret and debug: Given and already trained Deep Neural Net, and a set of test inputs, how can we gain insight into how those inputs interact with different layers of the neural network? Furthermore, can we characterize a given deep neural network based on it's observed behavior on different inputs? In this paper we propose a novel factorization based approach on understanding how different deep neural networks operate. In our preliminary results, we identify fascinating patterns that link the factorization rank (typically used as a measure of interestingness in unsupervised data analysis) with how well or poorly the deep network has been trained. Finally, our proposed approach can help provide visual insights on how high-level. interpretable patterns of the network's input behave inside the hidden layers of the deep network.

Large linear systems play an important role in high-energy theory, appearing in amplitude bootstraps and during integral reduction. This paper introduces FiniteFieldSolve, a general-purpose toolkit for exactly solving large linear systems over the rationals. The solver interfaces directly with Mathematica, is straightforward to install, and seamlessly replaces Mathematica's native solvers. In testing, FiniteFieldSolve is approximately two orders of magnitude faster than Mathematica and uses an order of magnitude less memory. The package also compares favorably against other public solvers in FiniteFieldSolve's intended use cases. As the name of the package suggests, solutions are obtained via well-known finite field methods. These methods suffer from introducing an inordinate number of modulo (or integer division) operations with respect to different primes. By automatically recompiling itself for each prime, FiniteFieldSolve converts the division operations into much faster combinations of instructions, dramatically improving performance. The technique of compiling the prime can be applied to any finite field solver, where the time savings will be solver dependent. The operation of the package is illustrated through a detailed example of an amplitude bootstrap.

The escalating demand for long-context applications has intensified the necessity of extending the LLM context windows. Despite recent fine-tuning approaches successfully expanding context lengths, their high memory footprints, especially for activations, present a critical practical limitation. Current parameter-efficient fine-tuning methods prioritize reducing parameter update overhead over addressing activation memory constraints. Similarly, existing sparsity mechanisms improve computational efficiency but overlook activation memory optimization due to the phenomenon of Shadowy Activation. In this paper, we propose LeMo, the first LLM fine-tuning system that explores and exploits a new token-level sparsity mechanism inherent in long-context scenarios, termed Contextual Token Sparsity. LeMo minimizes redundant token involvement by assessing the informativeness of token embeddings while preserving model accuracy. Specifically, LeMo introduces three key techniques: (1) Token Elimination, dynamically identifying and excluding redundant tokens across varying inputs and layers. (2) Pattern Prediction, utilizing well-trained predictors to approximate token sparsity patterns with minimal overhead. (3) Kernel Optimization, employing permutation-free and segment-based strategies to boost system performance. We implement LeMo as an end-to-end fine-tuning system compatible with various LLM architectures and other optimization techniques. Comprehensive evaluations demonstrate that LeMo reduces memory consumption by up to 1.93x and achieves up to 1.36x speedups, outperforming state-of-the-art fine-tuning systems.

The rapid progress of large language models (LLMs) has advanced numerous applications, yet efficient single-batch inference remains vital for on-device intelligence. While FPGAs offer fine-grained data control and high energy efficiency, recent GPU optimizations have narrowed their advantage, especially under arithmetic-based computation. To overcome this, we leverage FPGAs' abundant on-chip memory to shift LLM inference from arithmetic- to memory-based computation through table lookups. We present LUT-LLM, the first FPGA accelerator enabling 1B+ LLM inference via vector-quantized memory operations. Our analysis identifies activation-weight co-quantization as the most effective scheme, supported by (1) bandwidth-aware parallel centroid search, (2) efficient 2D table lookups, and (3) a spatial-temporal hybrid design minimizing data caching. Implemented on an AMD V80 FPGA for a customized Qwen 3 1.7B model, LUT-LLM achieves 1.66x lower latency than AMD MI210 and 1.72x higher energy efficiency than NVIDIA A100, scaling to 32B models with 2.16x efficiency gain over A100.

We show that Large Language Model from Power Law Decoder Representations (PLDR-LLM) is a foundational model whose deductive outputs are invariant tensors up to a small perturbation. PLDR-LLM learns a singularity condition for the deductive outputs that enable the once-inferred energy-curvature tensor G_{LM} to replace the deep neural network of power law graph attention (PLGA) generating the deductive outputs at inference. We demonstrate that a cache for G_{LM} (G-cache) and KV-cache can be implemented in a straightforward manner to improve the inference time. The invariance and generalizable nature of deductive outputs is at a very high fidelity where deductive outputs have same RMSE and determinant values up to 15 decimal places after caching, and zero-shot benchmark scores remain unchanged. Ablation studies show that learned deductive outputs have distinct loss and accuracy characteristics from models pretrained with transferred, randomly initialized or identity tensors as a constant tensor operator and an LLM with scaled-dot product attention (SDPA) is a special case of PLDR-LLM where G_{LM} is predefined as identity. The observed invariance characteristic introduces a novel asymmetry between training and inference phases with caching. We outline observed common characteristics of the deductive outputs for the learned singularity condition. We provide an implementation of a training and inference framework for PLDR-LLM with KV-cache and G-cache.

Matrix-based preconditioned optimizers, such as Muon, have recently been shown to be more efficient than scalar-based optimizers for training large-scale neural networks, including large language models (LLMs). On the other hand, recent benchmarks on optimizers for LLM pre-training have demonstrated that variance-reduction techniques such as MARS can achieve substantial speedups over standard optimizers that do not employ variance reduction. In this paper, to achieve the best of both worlds, we introduce MARS-M, a new optimizer that integrates the variance reduction technique in MARS with Muon. Under standard regularity conditions, we prove that Muon-M converges to a first-order stationary point at a rate of mathcal{O}(T^{-1/3}), which improves upon mathcal{O}(T^{-1/4}) rate attained by Muon. Our empirical results on language modeling and computer vision tasks demonstrate that MARS-M consistently yields lower losses and improved performance across various downstream benchmarks. The implementation of MARS-M is available at https://github.com/AGI-Arena/MARS/MARS_M.

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.

Large Language Model (LLM) inference requires substantial computational resources, yet CPU-based inference remains essential for democratizing AI due to the widespread availability of CPUs compared to specialized accelerators. However, efficient LLM inference on CPUs faces two fundamental challenges: (1) existing CPU architectures struggle with low-precision arithmetic required by quantized models, where optimal bit precision varies across models and layers; and (2) the memory-bound nature of the token generation phase creates severe performance bottlenecks. To address these challenges, we propose SAIL (SRAM-Accelerated Inference of LLMs), a CPU-based inference solution that efficiently supports arbitrary bit precisions with minimal overhead. SAIL integrates three key innovations: First, we introduce Batched LUT-based General Matrix-Vector Multiplication (LUT-GEMV) with SRAM-based processing-in-memory, enabling high data reuse through lookup tables and reducing memory movement. Second, our Pattern-Aware LUT optimization identifies and exploits redundancy in input activation patterns, reducing computation cycles by 13.8\%. Third, we develop an in-memory type conversion algorithm that leverages PIM's parallelism for efficient de-/quantization operations, alleviating pressure on CPU's vector units. Our architecture requires only 2\% hardware overhead and a single new instruction, while maintaining dual functionality as both compute and storage units. Experimental evaluations using a modified gem5 simulator demonstrate that SAIL achieves up to 10.7x speedup and 19.9x higher tokens per dollar compared to ARM Neoverse-N1 CPU baselines, and up to 7.04x better cost efficiency than NVIDIA V100 GPUs, establishing a practical path for efficient CPU-based LLM inference.

The Karhunen--Loève transform (KLT) of the stationary AR(1) source is the classical optimality benchmark of transform coding. Its frequencies are roots of the transcendental equations of Ray and Driver, and the transform has therefore long been treated as an unstructured dense matrix. Rather than compute the exact KLT, the field turned to fast approximations of it, most famously the discrete cosine transform (DCT). This paper shows that the retreat was premature. The exact AR(1) KLT admits direct recursive factorizations, at every order Nge2 and every correlation coefficient ρin(0,1): at even orders, the transform reduces to a butterfly stage, two half-order copies of itself, and orthogonal corrections generated by rank-one boundary perturbations of the covariance; at odd orders, to two copies of the KLT of the one-sided prediction residual and an arrowhead stage absorbing the center sample. The residual KLT recurses through half order as well. All correction stages are Cauchy-structured, and applying them by the fast multipole method yields exact-KLT algorithms of O(Nlog N) complexity---the same order as the FFT and the fast sinusoidal transforms. The factorizations follow from the covariance matrix by the Sherman--Morrison identity and the classical secular eigenvalue updates, and degenerate, as ρto1, to the known parity splittings of the DCT-II. As secondary results, the factorization constants are shown to be algebraic functions of ρ---in radicals for all Nle8---and complete exact KLT modules for N=2,dots,8 are given.

Diffusion Large Language Models (DLLMs) offer a compelling alternative to Auto-Regressive models, but their deployment is constrained by high decoding cost. In this work, we identify a key inefficiency in DLLM decoding: while computation is parallelized over token blocks, only a small subset of tokens is decodable at each diffusion step, causing most compute to be wasted on non-decodable tokens. We further observe a strong correlation between attention-derived token importance and token-wise decoding probability. Based on this insight, we propose FOCUS -- an inference system designed for DLLMs. By dynamically focusing computation on decodable tokens and evicting non-decodable ones on-the-fly, FOCUS increases the effective batch size, alleviating compute limitations and enabling scalable throughput. Empirical evaluations demonstrate that FOCUS achieves up to 3.52times throughput improvement over the production-grade engine LMDeploy, while preserving or improving generation quality across multiple benchmarks. The FOCUS system is publicly available on GitHub: https://github.com/sands-lab/FOCUS.

We present a reproducible, parameter-driven software workflow for optimizing approximate mutually unbiased basis (AMUB) configurations in arbitrary dimensions d using a Lie-algebra unitary parameterization. The workflow is designed for portable execution across CPU, Apple MPS, CUDA-capable GPU, and HPC backends, using a Taylor-series matrix exponential layer as an accelerator compatibility pathway. As a dimension-six case study, we optimize unanchored configurations across 100 random seeds for basis counts n = 3, 4, 5, 6 in complex128 and complex64 arithmetic. The workflow recovers exact three-basis configurations, identifies a recurrent four-basis partial-exact hub-and-triangle structure, and finds no near-exact pairs for n = 5 or n = 6 in the reported campaigns under the primary tolerance. As a hardware-execution check, we embed the representative d = 6, n = 4 transition unitaries into three-qubit 8x8 unitaries and execute the resulting circuits on the 156-qubit Heron processor ibm-marrakesh using subspace post-selection. The measured QPU pairwise losses are dominated by a hardware and compilation noise floor of approximately 0.02-0.08, associated with compiled circuits averaging 37 native CZ gates, which obscures the distinction between classically near-exact and defective pairs. The results provide a reproducible computational framework for exploring AMUB landscapes, together with an initial assessment of the challenges involved in executing optimized dimension-six unitaries on current quantum hardware.

We present version 3 of Kira, a Feynman integral reduction program for high-precision calculations in quantum field theory and gravitational-wave physics. Building on previous versions, Kira 3 introduces optimized seeding and equation selection algorithms, significantly improving performance for multi-loop and multi-scale problems. New features include convenient numerical sampling, symbolic integration-by-parts reductions, and support for user-defined additional relations. We demonstrate its capabilities through benchmarks on two- and three-loop topologies, showcasing up to two orders of magnitude improvement over Kira 2.3. Kira 3 is publicly available and poised to support ambitious projects in particle physics and beyond.

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a preconditioning procedure that accelerates Newton-Schulz convergence and reduces its computational cost. We evaluate its impact and show that the overhead of our preconditioning can be made negligible. Furthermore, the faster convergence it enables allows us to remove one iteration out of the usual five without degrading approximation quality. Our publicly available implementation achieves up to a 2.8x speedup in the Newton-Schulz approximation. We also show that this has a direct impact on end-to-end training runtime with 5-10% improvement in realistic training scenarios across two efficiency-focused tasks. On challenging language or vision tasks, we validate that our method maintains equal or superior model performance while improving runtime. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Our code is publicly available on github.

In this paper, we establish a connection between the parameterization of flow-based and energy-based generative models, and present a new flow-based modeling approach called energy-based normalizing flow (EBFlow). We demonstrate that by optimizing EBFlow with score-matching objectives, the computation of Jacobian determinants for linear transformations can be entirely bypassed. This feature enables the use of arbitrary linear layers in the construction of flow-based models without increasing the computational time complexity of each training iteration from O(D^2L) to O(D^3L) for an L-layered model that accepts D-dimensional inputs. This makes the training of EBFlow more efficient than the commonly-adopted maximum likelihood training method. In addition to the reduction in runtime, we enhance the training stability and empirical performance of EBFlow through a number of techniques developed based on our analysis of the score-matching methods. The experimental results demonstrate that our approach achieves a significant speedup compared to maximum likelihood estimation while outperforming prior methods with a noticeable margin in terms of negative log-likelihood (NLL).

Parallel decoding methods such as Jacobi decoding show promise for more efficient LLM inference as it breaks the sequential nature of the LLM decoding process and transforms it into parallelizable computation. However, in practice, it achieves little speedup compared to traditional autoregressive (AR) decoding, primarily because Jacobi decoding seldom accurately predicts more than one token in a single fixed-point iteration step. To address this, we develop a new approach aimed at realizing fast convergence from any state to the fixed point on a Jacobi trajectory. This is accomplished by refining the target LLM to consistently predict the fixed point given any state as input. Extensive experiments demonstrate the effectiveness of our method, showing 2.4times to 3.4times improvements in generation speed while preserving generation quality across both domain-specific and open-domain benchmarks.

Factorisation-based Models (FMs), such as DistMult, have enjoyed enduring success for Knowledge Graph Completion (KGC) tasks, often outperforming Graph Neural Networks (GNNs). However, unlike GNNs, FMs struggle to incorporate node features and generalise to unseen nodes in inductive settings. Our work bridges the gap between FMs and GNNs by proposing ReFactor GNNs. This new architecture draws upon both modelling paradigms, which previously were largely thought of as disjoint. Concretely, using a message-passing formalism, we show how FMs can be cast as GNNs by reformulating the gradient descent procedure as message-passing operations, which forms the basis of our ReFactor GNNs. Across a multitude of well-established KGC benchmarks, our ReFactor GNNs achieve comparable transductive performance to FMs, and state-of-the-art inductive performance while using an order of magnitude fewer parameters.

Semi-structured 2:4 sparsity is widely supported by modern accelerators, providing up to a 2x theoretical speedup. However, its strict 50% sparsity constraint often causes non-negligible accuracy degradation under post-training pruning. Meanwhile, existing relaxed sparsity formats either require specialized compiler support or introduce runtime overheads that limit end-to-end speedup. We propose Spense, a practical hybrid sparse-dense format that splits each weight matrix into a 2:4 sparse region and a dense region. This design relaxes the effective sparsity constraint while remaining compatible with existing high-performance sparse and dense GEMM libraries, avoiding both custom compiler support and input activation expansion. Building on this format, we introduce SpenseGPT, a one-shot post-training pruning method that produces sparse and dense regions. Notably, we show that selecting the right dense regions is important, and we devise two different strategies to choose them. Experiments on Qwen3-32B and Seed-OSS-36B demonstrate that our method achieves up to 1.2x end-to-end decoding speedup on B200 GPUs with FP8 precision, while preserving accuracy. To the best of our knowledge, this is the first one-shot pruning demonstration of real-world end-to-end LLM decoding speedup from semi-structured sparse tensor cores on recent GPUs such as B200s, while maintaining model quality.

Mathematical equations have been unreasonably effective in describing complex natural phenomena across various scientific disciplines. However, discovering such insightful equations from data presents significant challenges due to the necessity of navigating extremely high-dimensional combinatorial and nonlinear hypothesis spaces. Traditional methods of equation discovery largely focus on extracting equations from data alone, often neglecting the rich domain-specific prior knowledge that scientists typically depend on. To bridge this gap, we introduce LLM-SR, a novel approach that leverages the extensive scientific knowledge and robust code generation capabilities of Large Language Models (LLMs) to discover scientific equations from data in an efficient manner. Specifically, LLM-SR treats equations as programs with mathematical operators and combines LLMs' scientific priors with evolutionary search over equation programs. The LLM iteratively proposes new equation skeletons, drawing from its physical understanding, which are then optimized against data to estimate skeleton parameters. We demonstrate LLM-SR's effectiveness across three diverse scientific domains, where it discovers physically accurate equations that provide significantly better fits to in-domain and out-of-domain data compared to the well-established equation discovery baselines

Deep Neural Networks (DNNs) have been a large driver and enabler for AI breakthroughs in recent years. These models have been getting larger in their attempt to become more accurate and tackle new upcoming use-cases, including AR/VR and intelligent assistants. However, the training process of such large models is a costly and time-consuming process, which typically yields a single model to fit all targets. To mitigate this, various techniques have been proposed in the literature, including pruning, sparsification or quantization of the model weights and updates. While able to achieve high compression rates, they often incur computational overheads or accuracy penalties. Alternatively, factorization methods have been leveraged to incorporate low-rank compression in the training process. Similarly, such techniques (e.g.,~SVD) frequently rely on the computationally expensive decomposition of layers and are potentially sub-optimal for non-linear models, such as DNNs. In this work, we take a further step in designing efficient low-rank models and propose Maestro, a framework for trainable low-rank layers. Instead of regularly applying a priori decompositions such as SVD, the low-rank structure is built into the training process through a generalized variant of Ordered Dropout. This method imposes an importance ordering via sampling on the decomposed DNN structure. Our theoretical analysis demonstrates that our method recovers the SVD decomposition of linear mapping on uniformly distributed data and PCA for linear autoencoders. We further apply our technique on DNNs and empirically illustrate that Maestro enables the extraction of lower footprint models that preserve model performance while allowing for graceful accuracy-latency tradeoff for the deployment to devices of different capabilities.

Activation sparsity refers to the existence of considerable weakly-contributed elements among activation outputs. As a prevalent property of the models using the ReLU activation function, it has been proven a promising paradigm to boost model inference efficiency. Nevertheless, most large language models (LLMs) adopt activation functions without intrinsic activation sparsity (e.g., GELU and Swish). Some recent efforts have explored introducing ReLU or its variants as the substitutive activation function to help LLMs achieve activation sparsity and inference acceleration, but few can simultaneously obtain high sparsity and comparable model performance. This paper introduces an effective sparsification method named "ProSparse" to push LLMs for higher activation sparsity without decreasing model performance. Specifically, after substituting the activation function of LLMs with ReLU, ProSparse adopts progressive sparsity regularization with a factor smoothly increasing along sine curves in multiple stages. This can enhance activation sparsity and alleviate performance degradation by avoiding radical shifts in activation distribution. With ProSparse, we obtain high sparsity of 89.32% and 88.80% for LLaMA2-7B and LLaMA2-13B, respectively, achieving comparable performance to their original Swish-activated versions. Our inference acceleration experiments further demonstrate the practical acceleration brought by higher activation sparsity.

For the case of nonlinear second-order differential equations with a constant coefficient of the first derivative term and polynomial nonlinearities, the factorization conditions of Rosu and Cornejo-Perez are approached in two ways: (i) by commuting the subindices of the factorization functions in the two factorization conditions and (ii) by leaving invariant only the first factorization condition achieved by using monomials or polynomial sequences. For the first case, the factorization brackets commute and the generated equations are only equations of Ermakov-Pinney type. The second modification is non commuting, leading to nonlinear equations with different nonlinear force terms, but the same first-order part as the initially factored equation. It is illustrated for monomials with the examples of the generalized Fisher and FitzHugh-Nagumo initial equations. A polynomial sequence example is also included.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

In this article, we focus on decomposing latent representations in generative adversarial networks or learned feature representations in deep autoencoders into semantically controllable factors in a semisupervised manner, without modifying the original trained models. Particularly, we propose factors' decomposer-entangler network (FDEN) that learns to decompose a latent representation into mutually independent factors. Given a latent representation, the proposed framework draws a set of interpretable factors, each aligned to independent factors of variations by minimizing their total correlation in an information-theoretic means. As a plug-in method, we have applied our proposed FDEN to the existing networks of adversarially learned inference and pioneer network and performed computer vision tasks of image-to-image translation in semantic ways, e.g., changing styles, while keeping the identity of a subject, and object classification in a few-shot learning scheme. We have also validated the effectiveness of the proposed method with various ablation studies in the qualitative, quantitative, and statistical examination.

We introduce WriteSAE, the first sparse autoencoder that decomposes and edits the matrix cache write of state-space and hybrid recurrent language models, where residual SAEs cannot reach. Existing SAEs read residual streams, but Gated DeltaNet, Mamba-2, and RWKV-7 write to a d_k times d_v cache through rank-1 updates k_t v_t^top that no vector atom can replace. WriteSAE factors each decoder atom into the native write shape, exposes a closed form for the per-token logit shift, and trains under matched Frobenius norm so atoms swap one cache slot at a time. Atom substitution beats matched-norm ablation on 92.4% of n=4{,}851 firings at Qwen3.5-0.8B L9 H4, the 87-atom population test holds at 89.8%, the closed form predicts measured effects at R^2=0.98, and Mamba-2-370M substitutes at 88.1% over 2,500 firings. Sustained three-position installs at 3times lift midrank target-in-continuation from 33.3% to 100% under greedy decoding, the first behavioral install at the matrix-recurrent write site.

The Mixture-of-Experts (MoE) architecture has become a predominant paradigm for scaling large language models (LLMs). Despite offering strong performance and computational efficiency, large MoE-based LLMs like DeepSeek-V3-0324 and Kimi-K2-Instruct present serious challenges due to substantial memory requirements in deployment. While recent works have explored MoE compression to address this issue, existing methods often suffer from considerable accuracy drops (e.g., 7-14% relatively) even at modest compression rates. This paper introduces a novel Mixture-of-Basis-Experts (MoBE) method that achieves model compression while incurring minimal accuracy drops. Specifically, each up/gate matrix in an expert is decomposed via a rank decomposition as W = AB, where matrix A is unique to each expert. The relatively larger matrix B is further re-parameterized as a linear combination of basis matrices {Bi} shared across all experts within a given MoE layer. The factorization is learned by minimizing the reconstruction error relative to the original weight matrices. Experiments demonstrate that MoBE achieves notably lower accuracy drops compared to prior works. For instance, MoBE can reduce the parameter counts of Qwen3-235B-A22B-2507, DeepSeek-V3-0324 (671B) and Kimi-K2-Instruct (1T) by 24%-30% with only 1%-2% accuracy drop (about 2% drops when measured relatively).

We study low-precision computation of C=AB with both factors quantized. We derive an exact finite-dimensional identity for the expected squared product error under independent, zero-mean entrywise errors with known variance fields; it holds exactly for non-overloading subtractive dither and for independent stochastic rounding, and we empirically assess deterministic round-to-nearest (RTN). Using the product-preserving equivalence AB=(AT)(T^{-1}B), we formulate contraction-gauge preconditioning: jointly choosing a factor representation and its sharing pattern before quantization. Preconditioning can reduce product error but may require extra transformed, quantized copies of the opposite operand: a shared transform needs one copy, a block-specific transform up to one per block. Within the bounded family of positive diagonal gauges (folds), a geometric program computes a globally optimal shared fold and a linear program decides whether the identity fold is already optimal. For other families we derive computable selection statistics -- tail index for scaling, profile spread for partitioning, coherence and weighted-Gram energy for rotations, slice-energy covariance for hierarchy depth -- with upper bounds for ranking heuristic candidates. Across twelve linear products from a trained three-block image classifier, median within-product rank correlations between dither-model predictions and deterministic-RTN errors are 0.937 at 8 bits and 0.918 at 4 bits. The GP fold cuts held-out product error over the identity fold by 18.0% (8-bit) and 20.5% (4-bit) in geometric mean, beats a SmoothQuant-style grid baseline at both precisions and on ten of twelve products, and lowers composed logit MSE by 15.4% and 26.4%. We thus provide exact stochastic product-error accounting, certified selection within the diagonal family, and a common objective for evaluating reusable transform candidates under RTN.

Large Language Models (LLMs) have grown rapidly in size, creating significant challenges for efficient deployment on resource-constrained hardware. In this paper, we introduce Dynamic-Length Float (DFloat11), a lossless compression framework that reduces LLM size by 30% while preserving outputs that are bit-for-bit identical to the original model. DFloat11 is motivated by the low entropy in the BFloat16 weight representation of LLMs, which reveals significant inefficiency in existing storage format. By applying entropy coding, DFloat11 assigns dynamic-length encodings to weights based on frequency, achieving near information-optimal compression without any loss of precision. To facilitate efficient inference with dynamic-length encodings, we develop a custom GPU kernel for fast online decompression. Our design incorporates the following: (i) decomposition of memory-intensive lookup tables (LUTs) into compact LUTs that fit in GPU SRAM, (ii) a two-phase kernel for coordinating thread read/write positions using lightweight auxiliary variables, and (iii) transformer-block-level decompression to minimize latency. Experiments on recent models, including Llama-3.1, Qwen-2.5, and Gemma-3, validates our hypothesis that DFloat11 achieves around 30% model size reduction while preserving bit-for-bit exact outputs. Compared to a potential alternative of offloading parts of an uncompressed model to the CPU to meet memory constraints, DFloat11 achieves 1.9-38.8x higher throughput in token generation. With a fixed GPU memory budget, DFloat11 enables 5.3-13.17x longer context lengths than uncompressed models. Notably, our method enables lossless inference of Llama-3.1-405B, an 810GB model, on a single node equipped with 8x80GB GPUs. Our code and models are available at https://github.com/LeanModels/DFloat11.

Mathematical problem-solving is a key field in artificial intelligence (AI) and a critical benchmark for evaluating the capabilities of large language models (LLMs). While extensive research has focused on mathematical problem-solving, most existing work and datasets concentrate on computational tasks, leaving gaps in areas like mathematical analysis, which demands rigorous proofs and formal reasoning. We developed the DEMI-MathAnalysis dataset, comprising proof-based problems from mathematical analysis topics such as Sequences and Limits, Infinite Series, and Convex Functions. We also designed a guiding framework to rigorously enhance LLMs' ability to solve these problems. Through fine-tuning LLMs on this dataset and employing our framework, we observed significant improvements in their capability to generate logical, complete, and elegant proofs. This work addresses critical gaps in mathematical reasoning and contributes to advancing trustworthy AI capable of handling formalized mathematical language. The code is publicly accessible at LLMs for Mathematical Analysis.

We provide a rigorous analysis of implicit regularization in an overparametrized tensor factorization problem beyond the lazy training regime. For matrix factorization problems, this phenomenon has been studied in a number of works. A particular challenge has been to design universal initialization strategies which provably lead to implicit regularization in gradient-descent methods. At the same time, it has been argued by Cohen et. al. 2016 that more general classes of neural networks can be captured by considering tensor factorizations. However, in the tensor case, implicit regularization has only been rigorously established for gradient flow or in the lazy training regime. In this paper, we prove the first tensor result of its kind for gradient descent rather than gradient flow. We focus on the tubal tensor product and the associated notion of low tubal rank, encouraged by the relevance of this model for image data. We establish that gradient descent in an overparametrized tensor factorization model with a small random initialization exhibits an implicit bias towards solutions of low tubal rank. Our theoretical findings are illustrated in an extensive set of numerical simulations show-casing the dynamics predicted by our theory as well as the crucial role of using a small random initialization.

Sparse Autoencoders (SAEs) have demonstrated significant promise in interpreting the hidden states of language models by decomposing them into interpretable latent directions. However, training SAEs at scale remains challenging, especially when large dictionary sizes are used. While decoders can leverage sparse-aware kernels for efficiency, encoders still require computationally intensive linear operations with large output dimensions. To address this, we propose KronSAE, a novel architecture that factorizes the latent representation via Kronecker product decomposition, drastically reducing memory and computational overhead. Furthermore, we introduce mAND, a differentiable activation function approximating the binary AND operation, which improves interpretability and performance in our factorized framework.

Differentiable sparse linear algebra is foundational for scientific machine learning, yet PyTorch lacks a unified library for it: torch.sparse provides only low-level kernels and a non-differentiable, CPU-only spsolve, and torch.linalg is dense-only. We present torch-sla, an open-source library that fills this gap. It exposes a single autograd-aware API for direct, iterative, nonlinear, and eigenvalue solvers across five interchangeable backends -- SciPy and Eigen on CPU, cuDSS, CuPy, and a PyTorch-native iterative solver on GPU -- with automatic dispatch by device and problem size. The library further supports batched solves over shared or distinct sparsity patterns and distributed multi-GPU execution via domain decomposition with halo exchange. These capabilities are made scalable by an O(1)-graph adjoint differentiation framework and an autograd-compatible distributed halo-exchange layer. The library is available at https://www.torchsla.com/.

Large language models (LLMs) have revolutionized Natural Language Processing (NLP), but their size creates computational bottlenecks. We introduce a novel approach to create accurate, sparse foundational versions of performant LLMs that achieve full accuracy recovery for fine-tuning tasks at up to 70% sparsity. We achieve this for the LLaMA-2 7B model by combining the SparseGPT one-shot pruning method and sparse pretraining of those models on a subset of the SlimPajama dataset mixed with a Python subset of The Stack dataset. We exhibit training acceleration due to sparsity on Cerebras CS-3 chips that closely matches theoretical scaling. In addition, we establish inference acceleration of up to 3x on CPUs by utilizing Neural Magic's DeepSparse engine and 1.7x on GPUs through Neural Magic's nm-vllm engine. The above gains are realized via sparsity alone, thus enabling further gains through additional use of quantization. Specifically, we show a total speedup on CPUs for sparse-quantized LLaMA models of up to 8.6x. We demonstrate these results across diverse, challenging tasks, including chat, instruction following, code generation, arithmetic reasoning, and summarization to prove their generality. This work paves the way for rapidly creating smaller and faster LLMs without sacrificing accuracy.

Large Language Models (LLMs) such as ChatGPT and LlaMA are advancing rapidly in generative Artificial Intelligence (AI), but their immense size poses significant challenges, such as huge training and inference costs, substantial energy demands, and limitations for on-site deployment. Traditional compression methods such as pruning, distillation, and low-rank approximation focus on reducing the effective number of neurons in the network, while quantization focuses on reducing the numerical precision of individual weights to reduce the model size while keeping the number of neurons fixed. While these compression methods have been relatively successful in practice, there is no compelling reason to believe that truncating the number of neurons is an optimal strategy. In this context, this paper introduces CompactifAI, an innovative LLM compression approach using quantum-inspired Tensor Networks that focuses on the model's correlation space instead, allowing for a more controlled, refined and interpretable model compression. Our method is versatile and can be implemented with - or on top of - other compression techniques. As a benchmark, we demonstrate that a combination of CompactifAI with quantization allows to reduce a 93% the memory size of LlaMA 7B, reducing also 70% the number of parameters, accelerating 50% the training and 25% the inference times of the model, and just with a small accuracy drop of 2% - 3%, going much beyond of what is achievable today by other compression techniques. Our methods also allow to perform a refined layer sensitivity profiling, showing that deeper layers tend to be more suitable for tensor network compression, which is compatible with recent observations on the ineffectiveness of those layers for LLM performance. Our results imply that standard LLMs are, in fact, heavily overparametrized, and do not need to be large at all.

Low-Rank Adaptation (LoRA) reparameterizes a weight update as a product of two low-rank factors, but the Jacobian J_{G} of the generator mapping the factors to the weight matrix is rank-deficient, so the factor-space preconditioner J_{G}^* {F}_t J_{G} induced by any {W}-space preconditioner {F}_t is singular, and consequently the standard chain rule cannot be uniquely inverted to map a preconditioned {W}-space direction back to a factor-space update. We cast existing LoRA optimizers in a unified framework parameterized by two choices: (i) which invertible surrogate for J_{G}^* {F}_t J_{G} to use, and (ii) which {F}_t on {W} to use. Existing methods occupy four families along these axes: factor-space adaptive updates, block-diagonal surrogates for J_{G}^* J_{G}, Frobenius-residual pseudoinverse methods, and Riemannian manifold constraint. Within this design space, a gradient-statistics-aware {F}_t paired with a closed-form factor-space solve at {O}((m+n)r) memory remains underexplored. We propose AdaPreLoRA, which fills this gap by adopting the Adafactor diagonal Kronecker preconditioner {H}_t on {W} and selecting from the resulting factor-space solution family the element minimizing an {H}_t-weighted imbalance between the two factor contributions; by construction, the resulting factor update is the closest LoRA approximation to the preconditioned {W}-space direction under the {H}_t-weighted norm. Across GPT-2 (E2E), Mistral-7B and Qwen2-7B (GLUE, ARC, GSM8K), and diffusion-model personalization, AdaPreLoRA is competitive with or improves over a representative set of LoRA optimizers while keeping peak GPU memory at the LoRA optimizer level.

In statistical relational learning, the link prediction problem is key to automatically understand the structure of large knowledge bases. As in previous studies, we propose to solve this problem through latent factorization. However, here we make use of complex valued embeddings. The composition of complex embeddings can handle a large variety of binary relations, among them symmetric and antisymmetric relations. Compared to state-of-the-art models such as Neural Tensor Network and Holographic Embeddings, our approach based on complex embeddings is arguably simpler, as it only uses the Hermitian dot product, the complex counterpart of the standard dot product between real vectors. Our approach is scalable to large datasets as it remains linear in both space and time, while consistently outperforming alternative approaches on standard link prediction benchmarks.

Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine within the Muon algorithm for training deep neural networks. However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize high throughput over high precision. We introduce Polar Express, a new method for computing the polar decomposition. Like Newton-Schulz and other classical polynomial methods, our approach uses only matrix-matrix multiplications, making it very efficient on GPUs. Inspired by earlier work of Chen & Chow and Nakatsukasa & Freund, Polar Express adapts the update rule at each iteration by solving a minimax optimization problem. We prove that this strategy minimizes error in a worst-case sense, allowing Polar Express to converge as rapidly as possible both in the early iterations and asymptotically. We also address finite-precision issues, making it practical to use in bfloat16. When integrated into the Muon training framework, our method leads to consistent improvements in validation loss when training a GPT-2 model on one billion tokens from the FineWeb dataset, outperforming recent alternatives across a range of learning rates.

Graph neural networks that model 3D data, such as point clouds or atoms, are typically desired to be SO(3) equivariant, i.e., equivariant to 3D rotations. Unfortunately equivariant convolutions, which are a fundamental operation for equivariant networks, increase significantly in computational complexity as higher-order tensors are used. In this paper, we address this issue by reducing the SO(3) convolutions or tensor products to mathematically equivalent convolutions in SO(2) . This is accomplished by aligning the node embeddings' primary axis with the edge vectors, which sparsifies the tensor product and reduces the computational complexity from O(L^6) to O(L^3), where L is the degree of the representation. We demonstrate the potential implications of this improvement by proposing the Equivariant Spherical Channel Network (eSCN), a graph neural network utilizing our novel approach to equivariant convolutions, which achieves state-of-the-art results on the large-scale OC-20 and OC-22 datasets.

Large Language Models (LLMs) prompted to generate chain-of-thought (CoT) exhibit impressive reasoning capabilities. Recent attempts at prompt decomposition toward solving complex, multi-step reasoning problems depend on the ability of the LLM to simultaneously decompose and solve the problem. A significant disadvantage is that foundational LLMs are typically not available for fine-tuning, making adaptation computationally prohibitive. We believe (and demonstrate) that problem decomposition and solution generation are distinct capabilites, better addressed in separate modules, than by one monolithic LLM. We introduce DaSLaM, which uses a decomposition generator to decompose complex problems into subproblems that require fewer reasoning steps. These subproblems are answered by a solver. We use a relatively small (13B parameters) LM as the decomposition generator, which we train using policy gradient optimization to interact with a solver LM (regarded as black-box) and guide it through subproblems, thereby rendering our method solver-agnostic. Evaluation on multiple different reasoning datasets reveal that with our method, a 175 billion parameter LM (text-davinci-003) can produce competitive or even better performance, compared to its orders-of-magnitude larger successor, GPT-4. Additionally, we show that DaSLaM is not limited by the solver's capabilities as a function of scale; e.g., solver LMs with diverse sizes give significant performance improvement with our solver-agnostic decomposition technique. Exhaustive ablation studies evince the superiority of our modular finetuning technique over exorbitantly large decomposer LLMs, based on prompting alone.

Diffusion-based large language models (dLLMs) have emerged as a promising alternative to autoregressive (AR) LLMs, leveraging denoising-based generation to enable inherent parallelism. Even more and more open-sourced dLLM models emerge, yet their widespread adoption remains constrained by the lack of a standardized and efficient inference framework. We present dInfer, an efficient and extensible framework for dLLM inference. dInfer decomposes the inference pipeline into four modular components--model, diffusion iteration manager, decoding strategy, and KV-cache manager--and integrates novel algorithms for each component alongside system-level optimizations. Through this combination of algorithmic innovations and system enhancements, dInfer achieves substantial efficiency gains without compromising output quality on LLaDA-MoE. At batch size 1, it surpasses 1,100 tokens per second on HumanEval and averages over 800 tokens per second across six benchmarks on 8times H800 GPUs. Compared to prior systems, dInfer delivers a 10times speedup over Fast-dLLM while maintaining similar model performance. Even compared to the AR model (with a comparable number of activation parameters and performance) QWen2.5-3B, which is highly optimized with the latest vLLM inference engine, dInfer still delivers a 2-3times speedup. The implementation of dInfer is open-sourced at https://github.com/inclusionAI/dInfer.

The deployment of large language models (LLMs) is often constrained by memory bandwidth, where the primary bottleneck is the cost of transferring model parameters from the GPU's global memory to its registers. When coupled with custom kernels that fuse the dequantization and matmul operations, weight-only quantization can thus enable faster inference by reducing the amount of memory movement. However, developing high-performance kernels for weight-quantized LLMs presents substantial challenges, especially when the weights are compressed to non-evenly-divisible bit widths (e.g., 3 bits) with non-uniform, lookup table (LUT) quantization. This paper describes FLUTE, a flexible lookup table engine for LUT-quantized LLMs, which uses offline restructuring of the quantized weight matrix to minimize bit manipulations associated with unpacking, and vectorization and duplication of the lookup table to mitigate shared memory bandwidth constraints. At batch sizes < 32 and quantization group size of 128 (typical in LLM inference), the FLUTE kernel can be 2-4x faster than existing GEMM kernels. As an application of FLUTE, we explore a simple extension to lookup table-based NormalFloat quantization and apply it to quantize LLaMA3 to various configurations, obtaining competitive quantization performance against strong baselines while obtaining an end-to-end throughput increase of 1.5 to 2 times.

Large Language Models (LLMs) present an intriguing avenue for exploration in the field of formal theorem proving. Nevertheless, their full potential, particularly concerning the mitigation of hallucinations and refinement through prover error messages, remains an area that has yet to be thoroughly investigated. To enhance the effectiveness of LLMs in the field, we introduce the Lyra, a new framework that employs two distinct correction mechanisms: Tool Correction (TC) and Conjecture Correction (CC). To implement Tool Correction in the post-processing of formal proofs, we leverage prior knowledge to utilize predefined prover tools (e.g., Sledgehammer) for guiding the replacement of incorrect tools. Tool Correction significantly contributes to mitigating hallucinations, thereby improving the overall accuracy of the proof. In addition, we introduce Conjecture Correction, an error feedback mechanism designed to interact with prover to refine formal proof conjectures with prover error messages. Compared to the previous refinement framework, the proposed Conjecture Correction refines generation with instruction but does not collect paired (generation, error & refinement) prompts. Our method has achieved state-of-the-art (SOTA) performance on both miniF2F validation (48.0% -> 55.3%) and test (45.5% -> 51.2%). We also present 3 IMO problems solved by Lyra. We believe Tool Correction (post-process for hallucination mitigation) and Conjecture Correction (subgoal adjustment from interaction with environment) could provide a promising avenue for future research in this field.

Synthetic data generation has become a cornerstone for advancing large language models. However, the lack of the quantitative analysis for error tolerance became a critical bottleneck. Consequently, current filtering strategies fluctuate between two extremes: they are either overly aggressive, risking the exclusion of potentially valuable samples, or overly permissive, failing to eliminate erroneous samples effectively. To bridge this gap, this paper introduces Atomic Tree Operation Modeling (ATOM), a framework that decomposes data into functional units (f(x)rightarrow y). ATOM distinguishes benign Operand x perturbations from fatal Operator f perturbations. The former are needlessly discarded by aggressive filtering, while the latter slip through permissive filtering. Our experiments reveal a double dissociation: models are robust to operand perturbations but collapse under operator perturbations. By prioritizing operator over aggressive operand precision, our ATOM-synthesized data outperforms rigorous baselines (e.g., +3.1% gain over LIMA), suggesting that operator diversity matters more than operand precision. Our code is available at https://github.com/Lut-hub/ATOM.

Sparse matrix factorization is the problem of approximating a matrix Z by a product of J sparse factors X^{(J)} X^{(J-1)} ldots X^{(1)}. This paper focuses on identifiability issues that appear in this problem, in view of better understanding under which sparsity constraints the problem is well-posed. We give conditions under which the problem of factorizing a matrix into two sparse factors admits a unique solution, up to unavoidable permutation and scaling equivalences. Our general framework considers an arbitrary family of prescribed sparsity patterns, allowing us to capture more structured notions of sparsity than simply the count of nonzero entries. These conditions are shown to be related to essential uniqueness of exact matrix decomposition into a sum of rank-one matrices, with structured sparsity constraints. In particular, in the case of fixed-support sparse matrix factorization, we give a general sufficient condition for identifiability based on rank-one matrix completability, and we derive from it a completion algorithm that can verify if this sufficient condition is satisfied, and recover the entries in the two sparse factors if this is the case. A companion paper further exploits these conditions to derive identifiability properties and theoretically sound factorization methods for multi-layer sparse matrix factorization with support constraints associated to some well-known fast transforms such as the Hadamard or the Discrete Fourier Transforms.

Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing and reasoning tasks. However, their performance in the foundational domain of arithmetic remains unsatisfactory. When dealing with arithmetic tasks, LLMs often memorize specific examples rather than learning the underlying computational logic, limiting their ability to generalize to new problems. In this paper, we propose a Composable Arithmetic Execution Framework (CAEF) that enables LLMs to learn to execute step-by-step computations by emulating Turing Machines, thereby gaining a genuine understanding of computational logic. Moreover, the proposed framework is highly scalable, allowing composing learned operators to significantly reduce the difficulty of learning complex operators. In our evaluation, CAEF achieves nearly 100% accuracy across seven common mathematical operations on the LLaMA 3.1-8B model, effectively supporting computations involving operands with up to 100 digits, a level where GPT-4o falls short noticeably in some settings.

The Muon optimizer has demonstrated promising performance in pre-training large language models through gradient (or momentum) orthogonalization. In this work, we propose a simple yet effective enhancement to Muon, namely Muon+, which introduces an additional normalization step after orthogonalization. We demonstrate the effectiveness of Muon+ through extensive pre-training experiments across a wide range of model scales and architectures. Our evaluation includes GPT-style models ranging from 130M to 774M parameters and LLaMA-style models ranging from 60M to 1B parameters. We comprehensively evaluate the effectiveness of Muon+ in the compute-optimal training regime and further extend the token-to-parameter (T2P) ratio to an industrial level of approx 200. Experimental results show that Muon+ provides a consistent boost on training and validation perplexity over Muon. We provide our code here: https://github.com/K1seki221/MuonPlus.

Sparse computation offers a compelling solution for the inference of Large Language Models (LLMs) in low-resource scenarios by dynamically skipping the computation of inactive neurons. While traditional approaches focus on ReLU-based LLMs, leveraging zeros in activation values, we broaden the scope of sparse LLMs beyond zero activation values. We introduce a general method that defines neuron activation through neuron output magnitudes and a tailored magnitude threshold, demonstrating that non-ReLU LLMs also exhibit sparse activation. To find the most efficient activation function for sparse computation, we propose a systematic framework to examine the sparsity of LLMs from three aspects: the trade-off between sparsity and performance, the predictivity of sparsity, and the hardware affinity. We conduct thorough experiments on LLMs utilizing different activation functions, including ReLU, SwiGLU, ReGLU, and ReLU^2. The results indicate that models employing ReLU^2 excel across all three evaluation aspects, highlighting its potential as an efficient activation function for sparse LLMs. We will release the code to facilitate future research.

Neural network pruning is a promising technique to mitigate the excessive computational and memory requirements of large language models (LLMs). Despite its promise, however, progress in this area has diminished, as conventional methods are seemingly unable to surpass moderate sparsity levels (50-60%) without severely degrading model accuracy. This work breaks through the current impasse, presenting a principled and effective method called Elsa, which achieves extreme sparsity levels of up to 90% while retaining high model fidelity. This is done by identifying several limitations in current practice, all of which can be traced back to their reliance on a surrogate objective formulation. Elsa tackles this issue directly and effectively via standard and well-established constrained optimization techniques based on ADMM. Our extensive experiments across a wide range of models and scales show that Elsa achieves substantial improvements over existing methods; e.g., it achieves 7.8times less perplexity than the best existing method on LLaMA-2-7B at 90% sparsity. Moreover, we show that Elsa remains stable even at extreme sparsity (e.g., 95\%), yielding up to times3.98 inference speedup and times7.80 memory compression over its dense counterpart. We also present Elsa_{-L}, a quantized variant that scales to extremely large models (27B), and establish its theoretical convergence guarantees.These results highlight meaningful progress in advancing the frontier of LLM sparsity, while promising that significant opportunities for further advancement may remain in directions that have so far attracted limited exploration.

Recently, the Muon optimizer based on matrix orthogonalization has demonstrated strong results in training small-scale language models, but the scalability to larger models has not been proven. We identify two crucial techniques for scaling up Muon: (1) adding weight decay and (2) carefully adjusting the per-parameter update scale. These techniques allow Muon to work out-of-the-box on large-scale training without the need of hyper-parameter tuning. Scaling law experiments indicate that Muon achieves sim!2times computational efficiency compared to AdamW with compute optimal training. Based on these improvements, we introduce Moonlight, a 3B/16B-parameter Mixture-of-Expert (MoE) model trained with 5.7T tokens using Muon. Our model improves the current Pareto frontier, achieving better performance with much fewer training FLOPs compared to prior models. We open-source our distributed Muon implementation that is memory optimal and communication efficient. We also release the pretrained, instruction-tuned, and intermediate checkpoints to support future research.

Under the conditional framework breaking the longstanding O(n log n) integer multiplication barrier (originating from OpenAI Preprint #109), the asymptotic time complexity exponent T(n) = O(n (log n)^{1-κ}) is governed by bounding the second moment E[X^2] < 1 of recursive branching processes over trilinear tensor carrier graphs. In this work, we present a constructive combinatorial optimization over the alternating carrier bipartite matching graphs of the Schönhage trilinear tensor decomposition at dimensions h=23 and h=25. Through an alternating augmenting path search, we uncover a non-trivial cluster-unlocking cascade that eliminates 185 carrier roles (95 roles in h=23 and 90 roles in h=25), reducing the global physical wire width by 377,890 wires from the reviewed baseline W = 177,284,805 down to W = 176,906,915. Evaluated over the exact 10^{-14} rational grid, this wire reduction certifies an improved second-moment coefficient AB = 5,164,059 / 125,000,000,000 and establishes the sharpened conditional exponent κ= 826,215,307 / 20,000,000,000,000 \approx 4.131076535 \times 10^{-5}, strictly surpassing the preceding record on this architecture (κ\approx 4.123864 \times 10^{-5}). All 47 polynomial constraints and 7 rational security margins are audited in exact Q-arithmetic with zero floating-point error, and the fundamental wire reduction identity is formally verified in the Lean 4 interactive theorem prover without unproven axioms.

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its theoretical understanding has lagged behind practice. In particular, practical implementations incorporate Nesterov momentum, compute the polar factor only approximately, and operate with stochastic gradients that may be heavy-tailed. We close this gap by developing a convergence theory for Muon with Nesterov momentum and inexact polar decomposition in non-convex matrix optimization under heavy-tailed noise. Our analysis builds on a unified framework for inexact polar decomposition that captures practical iterative approximations such as Newton-Schulz and quantifies how their errors propagate through the optimization dynamics. Under this framework, we establish an optimal iteration and sample complexity of O left(varepsilon^{-(3α-2){(α-1)}} right) for finding an varepsilon-stationary point, where αin(1,2] denotes the heavy-tail index. For the inexact-polar setting with σ_1=0, we also provide guarantees that do not require prior knowledge of α. We analyze a randomized low-rank polar decomposition that is substantially more efficient than full-space methods while remaining compatible with our theory. Numerical experiments further demonstrate the effectiveness of the proposed inexact and randomized variants.

Despite their success in many natural language tasks, solving math problems remains a significant challenge for large language models (LLMs). A large gap exists between LLMs' pass-at-one and pass-at-N performance in solving math problems, suggesting LLMs might be close to finding correct solutions, motivating our exploration of fine-tuning methods to unlock LLMs' performance. Using the challenging MATH dataset, we investigate three fine-tuning strategies: (1) solution fine-tuning, where we fine-tune to generate a detailed solution for a given math problem; (2) solution-cluster re-ranking, where the LLM is fine-tuned as a solution verifier/evaluator to choose among generated candidate solution clusters; (3) multi-task sequential fine-tuning, which integrates both solution generation and evaluation tasks together efficiently to enhance the LLM performance. With these methods, we present a thorough empirical study on a series of PaLM 2 models and find: (1) The quality and style of the step-by-step solutions used for fine-tuning can make a significant impact on the model performance; (2) While solution re-ranking and majority voting are both effective for improving the model performance when used separately, they can also be used together for an even greater performance boost; (3) Multi-task fine-tuning that sequentially separates the solution generation and evaluation tasks can offer improved performance compared with the solution fine-tuning baseline. Guided by these insights, we design a fine-tuning recipe that yields approximately 58.8% accuracy on the MATH dataset with fine-tuned PaLM 2-L models, an 11.2% accuracy improvement over the few-shot performance of pre-trained PaLM 2-L model with majority voting.

Recent work leverages Vision Foundation Models as image encoders to boost the generative performance of latent diffusion models (LDMs), as their semantic feature distributions are easy to learn. However, such semantic features often lack low-level information (\eg, color and texture), leading to degraded reconstruction fidelity, which has emerged as a primary bottleneck in further scaling LDMs. To address this limitation, we propose LV-RAE, a representation autoencoder that augments semantic features with missing low-level information, enabling high-fidelity reconstruction while remaining highly aligned with the semantic distribution. We further observe that the resulting high-dimensional, information-rich latent make decoders sensitive to latent perturbations, causing severe artifacts when decoding generated latent and consequently degrading generation quality. Our analysis suggests that this sensitivity primarily stems from excessive decoder responses along directions off the data manifold. Building on these insights, we propose fine-tuning the decoder to increase its robustness and smoothing the generated latent via controlled noise injection, thereby enhancing generation quality. Experiments demonstrate that LV-RAE significantly improves reconstruction fidelity while preserving the semantic abstraction and achieving strong generative quality. Our code is available at https://github.com/modyu-liu/LVRAE.

Recently, sequential recommendation has been adapted to the LLM paradigm to enjoy the power of LLMs. LLM-based methods usually formulate recommendation information into natural language and the model is trained to predict the next item in an auto-regressive manner. Despite their notable success, the substantial computational overhead of inference poses a significant obstacle to their real-world applicability. In this work, we endeavor to streamline existing LLM-based recommendation models and propose a simple yet highly effective model Lite-LLM4Rec. The primary goal of Lite-LLM4Rec is to achieve efficient inference for the sequential recommendation task. Lite-LLM4Rec circumvents the beam search decoding by using a straight item projection head for ranking scores generation. This design stems from our empirical observation that beam search decoding is ultimately unnecessary for sequential recommendations. Additionally, Lite-LLM4Rec introduces a hierarchical LLM structure tailored to efficiently handle the extensive contextual information associated with items, thereby reducing computational overhead while enjoying the capabilities of LLMs. Experiments on three publicly available datasets corroborate the effectiveness of Lite-LLM4Rec in both performance and inference efficiency (notably 46.8% performance improvement and 97.28% efficiency improvement on ML-1m) over existing LLM-based methods. Our implementations will be open sourced.

Low-Rank Adaptation (LoRA) has become the de facto parameter-efficient fine-tuning (PEFT) method for large language models (LLMs) by constraining weight updates to low-rank matrices. Recent works such as Tied-LoRA, VeRA, and VB-LoRA push efficiency further by introducing additional constraints to reduce the trainable parameter space. In this paper, we show that the parameter space reduction strategies employed by these LoRA variants can be formulated within a unified framework, Uni-LoRA, where the LoRA parameter space, flattened as a high-dimensional vector space R^D, can be reconstructed through a projection from a subspace R^d, with d ll D. We demonstrate that the fundamental difference among various LoRA methods lies in the choice of the projection matrix, P in R^{D times d}.Most existing LoRA variants rely on layer-wise or structure-specific projections that limit cross-layer parameter sharing, thereby compromising parameter efficiency. In light of this, we introduce an efficient and theoretically grounded projection matrix that is isometric, enabling global parameter sharing and reducing computation overhead. Furthermore, under the unified view of Uni-LoRA, this design requires only a single trainable vector to reconstruct LoRA parameters for the entire LLM - making Uni-LoRA both a unified framework and a "one-vector-only" solution. Extensive experiments on GLUE, mathematical reasoning, and instruction tuning benchmarks demonstrate that Uni-LoRA achieves state-of-the-art parameter efficiency while outperforming or matching prior approaches in predictive performance. Our code is available at https://github.com/KaiyangLi1992/Uni-LoRA.

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.

Deploying large language models (LLMs) on edge devices presents significant challenges due to the substantial computational overhead and memory requirements. Activation sparsification can mitigate these challenges by reducing the number of activated neurons during inference. Existing methods typically employ thresholding-based sparsification based on the statistics of activation tensors. However, these methods do not explicitly model the impact of activation sparsification on performance, leading to suboptimal performance degradation. To address this issue, this paper reformulates the activation sparsification problem by introducing a new objective that optimizes the sparsification decisions. Building on this reformulation, we propose CHESS, a general activation sparsification approach via CHannel-wise thrEsholding and Selective Sparsification. First, channel-wise thresholding assigns a unique threshold to each activation channel in the feed-forward network (FFN) layers. Then, selective sparsification involves applying thresholding-based activation sparsification to specific layers within the attention modules. Finally, we detail the implementation of sparse kernels to accelerate LLM inference. Experimental results demonstrate that the proposed CHESS achieves lower performance degradation over 8 downstream tasks while activating fewer parameters compared to existing methods, thus speeding up the LLM inference by up to 1.27x.

Optimizers that exploit the matrix structure of gradients are central to modern LLM pre-training, with two distinct frontiers: explicit Kronecker-factored preconditioning -- most recently KL-Shampoo, which estimates the preconditioner via KL divergence minimization -- and orthogonalization of the gradient momentum, exemplified by Muon and analyzed as steepest descent under the spectral norm. The two routes are typically developed in isolation. We make a structural observation about KL-Shampoo's Kronecker preconditioners: their eigenvalue spectra exhibit a spike-and-flat shape -- a few dominant eigenvalues followed by an approximately uniform tail -- across layers and training stages, holding exactly under a rank-ρ signal-plus-noise gradient model. We exploit this structure by restricting one of KL-Shampoo's Kronecker factors to a parametric family aligned with the spike-and-flat shape: full spectral structure on a tracked r-dimensional subspace, single shared eigenvalue across the remaining n-r directions. On these directions, we apply orthogonalization. An identity shows that this orthogonalization recovers the algebraic form of full KL-Shampoo's preconditioner. On four pre-training scales (GPT-2 124M / 350M, LLaMA 134M / 450M), Pro-KLShampoo consistently outperforms KL-Shampoo at every subspace rank we test in validation loss, peak per-GPU memory, and wallclock time to reach each loss level.

We introduce a fast low-rank factorization-based framework for compressing large language models that enables rapid compression of billion-parameter models without retraining. Unlike existing factorization-based approaches that optimize only on the original inputs, ignoring distribution shifts from upstream compression and thus propagating errors forward, or those that rely only on shifted inputs and risk drifting away from the original outputs, our approach accounts for both. Beyond individual layer compression, we further refine each transformer block end-to-end, minimizing block-level output distortion and allowing compressed layers to jointly compensate for accumulated errors. By anchoring each compressed layer to the original outputs while explicitly modeling input distribution shifts, our method finds a low-rank approximation that maintains functional equivalence with the original model. Experiments on large language models show that our method consistently outperforms existing SVD-based baselines across compression ratios, with the advantage becoming increasingly pronounced at aggressive compression budgets, where competing methods degrade substantially or collapse entirely, offering a practical solution for efficient, large-scale model deployment.

Activation sparsity is an intriguing property of deep neural networks that has been extensively studied in ReLU-based models, due to its advantages for efficiency, robustness, and interpretability. However, methods relying on exact zero activations do not directly apply to modern Large Language Models (LLMs), leading to fragmented, model-specific strategies for LLM activation sparsity and a gap in its general understanding. In this work, we introduce a general framework for evaluating sparsity robustness in contemporary LLMs and conduct a systematic investigation of this phenomenon in their feedforward~(FFN) layers. Our results uncover universal properties of activation sparsity across diverse model families and scales. Importantly, we observe that the potential for effective activation sparsity grows with model size, highlighting its increasing relevance as models scale. Furthermore, we present the first study of activation sparsity in diffusion-based LLMs. Overall, our work provides a comprehensive perspective and practical guidance for harnessing activation sparsity in LLM design and acceleration.

Large Language Models (LLMs) have exhibited exceptional performance across a broad range of tasks and domains. However, they still encounter difficulties in solving mathematical problems due to the rigorous and logical nature of mathematics. Previous studies have employed techniques such as supervised fine-tuning (SFT), prompt engineering, and search-based methods to improve the mathematical problem-solving abilities of LLMs. Despite these efforts, their performance remains suboptimal and demands substantial computational resources. To address this issue, we propose a novel approach, BEATS, to enhance mathematical problem-solving abilities. Our method leverages newly designed prompts that guide the model to iteratively rewrite, advance by one step, and generate answers based on previous steps. Additionally, we introduce a new back-verification technique that uses LLMs to validate the correctness of the generated answers. Furthermore, we employ a pruning tree search to optimize search time while achieving strong performance. Notably, our method improves Qwen2-7b-Instruct's score from 36.94 to 61.52, outperforming GPT4's 42.5 on the MATH benchmark.

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

Library learning compresses a given corpus of programs by extracting common structure from the corpus into reusable library functions. Prior work on library learning suffers from two limitations that prevent it from scaling to larger, more complex inputs. First, it explores too many candidate library functions that are not useful for compression. Second, it is not robust to syntactic variation in the input. We propose library learning modulo theory (LLMT), a new library learning algorithm that additionally takes as input an equational theory for a given problem domain. LLMT uses e-graphs and equality saturation to compactly represent the space of programs equivalent modulo the theory, and uses a novel e-graph anti-unification technique to find common patterns in the corpus more directly and efficiently. We implemented LLMT in a tool named BABBLE. Our evaluation shows that BABBLE achieves better compression orders of magnitude faster than the state of the art. We also provide a qualitative evaluation showing that BABBLE learns reusable functions on inputs previously out of reach for library learning.

Mathematical reasoning remains a significant challenge for Large Language Models (LLMs) due to hallucinations. When combined with formal proof assistants like Lean, these hallucinations can be eliminated through rigorous verification, making theorem proving reliable. However, even with formal verification, LLMs still struggle with long proofs and complex mathematical formalizations. While Lean with LLMs offers valuable assistance with retrieving lemmas, generating tactics, or even complete proofs, it lacks a crucial capability: providing a sense of proof progress. This limitation particularly impacts the overall development efficiency in large formalization projects. We introduce LeanProgress, a method that predicts the progress in the proof. Training and evaluating our models made on a large corpus of Lean proofs from Lean Workbook Plus and Mathlib4 and how many steps remain to complete it, we employ data preprocessing and balancing techniques to handle the skewed distribution of proof lengths. Our experiments show that LeanProgress achieves an overall prediction accuracy of 75.1\% in predicting the amount of progress and, hence, the remaining number of steps. When integrated into a best-first search framework using Reprover, our method shows a 3.8\% improvement on Mathlib4 compared to baseline performances of 41.2\%, particularly for longer proofs. These results demonstrate how proof progress prediction can enhance both automated and interactive theorem proving, enabling users to make more informed decisions about proof strategies.

Lagrangian Relaxation (LR) is a powerful technique for solving large-scale Mixed Integer Linear Programming (MILP), particularly those with decomposable structures, such as vehicle routing or unit commitment problems. By relaxing the coupling constraints, LR enables parallel subproblem solving and often yields tighter dual bounds than standard linear programming relaxations, which is crucial for efficient branch-and-bound pruning. While recent empirical work has shown promising results using machine learning to predict these multipliers, a theoretical understanding of such methods remains an open question. In this work, we bridge this gap by analyzing the problem of learning LR through the lens of Data-driven Algorithm Design, i.e., a statistical learning problem over a distribution of problem instances. Our contributions are as follows: first, we derive a generalization bound of O(s^{1.5}/N) for the learned multipliers, where s is the number of coupling constraints and N is the sample size. Second, we provide a minimax lower-bound of Ω(s/N), proving that a linear dependency is unavoidable. Third, we constructively close this theoretical gap by proving that Stochastic Gradient Ascent (SGA) with averaging achieves the minimax optimal rate Θ(s/N). Finally, we extend our framework to the learning-to-warm-start setting, proving that it achieves a fast, minimax-optimal rate of Θ(s/N) and establishing a theoretical advantage over direct multiplier prediction.

Large Language Models (LLMs) have demonstrated exceptional capabilities in various natural language tasks, often achieving performances that surpass those of humans. Despite these advancements, the domain of mathematics presents a distinctive challenge, primarily due to its specialized structure and the precision it demands. In this study, we adopted a two-step approach for investigating the proficiency of LLMs in answering mathematical questions. First, we employ the most effective LLMs, as identified by their performance on math question-answer benchmarks, to generate answers to 78 questions from the Math Stack Exchange (MSE). Second, a case analysis is conducted on the LLM that showed the highest performance, focusing on the quality and accuracy of its answers through manual evaluation. We found that GPT-4 performs best (nDCG of 0.48 and P@10 of 0.37) amongst existing LLMs fine-tuned for answering mathematics questions and outperforms the current best approach on ArqMATH3 Task1, considering P@10. Our Case analysis indicates that while the GPT-4 can generate relevant responses in certain instances, it does not consistently answer all questions accurately. This paper explores the current limitations of LLMs in navigating complex mathematical problem-solving. Through case analysis, we shed light on the gaps in LLM capabilities within mathematics, thereby setting the stage for future research and advancements in AI-driven mathematical reasoning. We make our code and findings publicly available for research: https://github.com/gipplab/LLM-Investig-MathStackExchange

Automated proving of polynomial inequalities is a fundamental challenge in automated mathematical reasoning, where rich algebraic structure and a rapidly growing certificate search space hinder scalability. Purely symbolic approaches provide strong guarantees but often scale poorly as the number of variables or the degree increases, due to expensive algebraic manipulations and rapidly growing intermediate expressions. In parallel, LLM-guided methods have made notable progress, particularly on competition-style inequalities with a small number of variables. To address the remaining scalability challenges, we propose NSPI, a neuro-symbolic framework that combines the complementary strengths of LLMs and symbolic computation for polynomial-inequality proving. Concretely, an LLM proposes a conjecture in the form of an approximate polynomial Sum-Of-Squares (SOS) decomposition; we refine it via symbolic computation to obtain an exact polynomial SOS representation, which directly proves the target inequality, and we further certify the proof in Lean, yielding an end-to-end pipeline from heuristic discovery to machine-checked proof. Experiments on challenging benchmarks involving polynomials with up to 10 variables demonstrate the effectiveness and scalability of the proposed method.

Large language models (LLMs) have demonstrated state-of-the-art performance across various tasks. However, the latency of inference and the large GPU memory consumption of LLMs restrict their deployment performance. Recently, there have been some efficient attempts to quantize LLMs, yet inference with large batch size or long sequence still has the issue of being compute-bound. Fine-grained quantization methods have showcased their proficiency in achieving low-bit quantization for LLMs, while requiring FP16 data type for linear layer computations, which is time-consuming when dealing with large batch size or long sequence. In this paper, we introduce a method called FlattenQuant, which significantly reduces the maximum value of the tensor by flattening the large channels in the tensor, to achieve low bit per-tensor quantization with minimal accuracy loss. Our experiments show that FlattenQuant can directly use 4 bits to achieve 48.29% of the linear layer calculation in LLMs, with the remaining layers using 8 bits. The 4-bit matrix multiplication introduced in the FlattenQuant method can effectively address the compute-bound caused by large matrix calculation. Our work achieves up to 2times speedup and 2.3times memory reduction for LLMs with negligible loss in accuracy.

Accelerating large language model (LLM) inference is critical for real-world deployments requiring high throughput and low latency. Contextual sparsity, where each token dynamically activates only a small subset of the model parameters, shows promise but does not scale to large batch sizes due to union of active neurons quickly approaching dense computation. We introduce Polar Sparsity, highlighting a key shift in sparsity importance from MLP to Attention layers as we scale batch size and sequence length. While MLP layers become more compute-efficient under batching, their sparsity vanishes. In contrast, attention becomes increasingly more expensive at scale, while their head sparsity remains stable and batch-invariant. We develop hardware-efficient, sparsity-aware GPU kernels for selective MLP and Attention computations, delivering up to \(2.2\times\) end-to-end speedups for models like OPT, LLaMA-2 \& 3, across various batch sizes and sequence lengths without compromising accuracy. To our knowledge, this is the first work to demonstrate that contextual sparsity can scale effectively to large batch sizes, delivering substantial inference acceleration with minimal changes, making Polar Sparsity practical for large-scale, high-throughput LLM deployment systems. Our code is available at: https://github.com/susavlsh10/Polar-Sparsity.

We have witnessed that strong LLMs like Qwen-Math, MiMo, and Phi-4 possess immense reasoning potential inherited from the pre-training stage. With reinforcement learning (RL), these models can improve dramatically on reasoning tasks. Recent studies have shown that even RL on a single problem can unleash these models' reasoning capabilities. However, RL is not only expensive but also unstable. Even one-shot RL requires hundreds of GPU hours. This raises a critical question: Is there a more efficient way to unleash the reasoning potential of these powerful base LLMs? In this work, we demonstrate that Critique Fine-Tuning (CFT) on only one problem can effectively unleash the reasoning potential of LLMs. Our method constructs critique data by collecting diverse model-generated solutions to a single problem and using teacher LLMs to provide detailed critiques. We fine-tune Qwen and Llama family models, ranging from 1.5B to 14B parameters, on the CFT data and observe significant performance gains across diverse reasoning tasks. For example, with just 5 GPU hours of training, Qwen-Math-7B-CFT show an average improvement of 15% on six math benchmarks and 16% on three logic reasoning benchmarks. These results are comparable to or even surpass the results from RL with 20x less compute. Ablation studies reveal the robustness of one-shot CFT across different prompt problems. These results highlight one-shot CFT as a simple, general, and compute-efficient approach to unleashing the reasoning capabilities of modern LLMs.

Multi-token generation has emerged as a promising paradigm for accelerating transformer-based large model inference. Recent efforts primarily explore diffusion Large Language Models (dLLMs) for parallel decoding to reduce inference latency. To achieve AR-level generation quality, many techniques adapt AR models into dLLMs to enable parallel decoding. However, they suffer from limited speedup compared to AR models due to a pretrain-to-posttrain mismatch. Specifically, the masked data distribution in post-training deviates significantly from the real-world data distribution seen during pretraining, and dLLMs rely on bidirectional attention, which conflicts with the causal prior learned during pretraining and hinders the integration of exact KV cache reuse. To address this, we introduce Jacobi Forcing, a progressive distillation paradigm where models are trained on their own generated parallel decoding trajectories, smoothly shifting AR models into efficient parallel decoders while preserving their pretrained causal inference property. The models trained under this paradigm, Jacobi Forcing Model, achieves 3.8x wall-clock speedup on coding and math benchmarks with minimal loss in performance. Based on Jacobi Forcing Models' trajectory characteristics, we introduce multi-block decoding with rejection recycling, which enables up to 4.5x higher token acceptance count per iteration and nearly 4.0x wall-clock speedup, effectively trading additional compute for lower inference latency. Our code is available at https://github.com/hao-ai-lab/JacobiForcing.

A persistent and fascinating problem at the high energy colliders are jets. Often trying to observe physics underlying the hard interactions at colliders requires experimental cuts in phase space, defining several jet or beam regions. QCD being a gauge theory that readily decays into infra-red modes, correlations between jet regions is almost inevitable, spoiling the predictivity of fixed order QCD calculations. One is faced with the task of calculating the evolution of a reduced density matrix, where successively less energetic (jet) regions are integrated out, to gain control of the calculation. I relate the decay rates governing the flow into the IR to an effective field theory expansion in soft jets, allowing a systematic and resummed calculation of these rates, while further relating them to physically observable features of the QCD cascade. To demonstrate the utility of the soft jet expansion, I present a factorization theorem for a soft subjet collinearly splitting in and out of a parent fat jet. Using the resummation properties of this factorization theorem, I elucidate the structure of the subleading non-global logs (encoding the jet correlations) in the hemisphere jet mass distribution, as well as give a collinear improvement of the leading order resummation equation, the BMS equation. I compare to other approaches to subleading resummation of NGLs, and find the collinear improvement of the leading order equation removes the need for kinematic-dependent corrections in the IR averaging procedure of the reduced density matrix, so that no further large logs can be generated in the IR. Finally I end with speculation about connections with collinear improvements of the NLO B-JIMWLK hierarchy for small-x resummation.

Optimization modeling via mixed-integer linear programming (MILP) is fundamental to industrial planning and scheduling, yet translating natural-language requirements into solver-executable models and maintaining them under evolving business rules remains highly expertise-intensive. While large language models (LLMs) offer promising avenues for automation, existing methods often suffer from low data efficiency, limited solver-level validity, and poor scalability to industrial-scale problems. To address these challenges, we present EvoOpt-LLM, a unified LLM-based framework supporting the full lifecycle of industrial optimization modeling, including automated model construction, dynamic business-constraint injection, and end-to-end variable pruning. Built on a 7B-parameter LLM and adapted via parameter-efficient LoRA fine-tuning, EvoOpt-LLM achieves a generation rate of 91% and an executability rate of 65.9% with only 3,000 training samples, with critical performance gains emerging under 1,500 samples. The constraint injection module reliably augments existing MILP models while preserving original objectives, and the variable pruning module enhances computational efficiency, achieving an F1 score of ~0.56 on medium-sized LP models with only 400 samples. EvoOpt-LLM demonstrates a practical, data-efficient approach to industrial optimization modeling, reducing reliance on expert intervention while improving adaptability and solver efficiency.

Modern Large Language Models (LLMs) are composed of matrices with billions of elements, making their storage and processing quite demanding in terms of computational resources and memory usage. Being significantly large, such matrices can often be expressed in low-rank format with potential to relax resource requirements. Unlike prior works which focus on developing novel matrix decomposition algorithms, in this work we first study the emergence of low-rank structures across matrices within different layers of LLMs and establish a consequential relationship between the gradient dynamics and emerging low-rank expressiveness of matrices. Our findings reveal that different layers exhibit varying levels of converged low-rank structure, necessitating a non-uniform rank reduction across them to minimize performance drop due to compression. In view of that, we present Weight Low-Rank Projection (WeLore) that unifies weight compression and memory-efficient fine-tuning as ONE, in a data-agnostic and one-shot way. WeLore capitalizes the heavy-tail distribution of singular values to identify a suitable rank reduction ratio for matrices within LLMs. Going beyond only as a compression technique, WeLore categorizes weight matrices into Low-rank Components (LRCs) and Non-Low-rank Components (N-LRCs) based on their ability to express themselves as low-rank. Our gradient perspective and extensive experiments illustrate that LRCs tend to have better finetuning capabilities and can closely mimic (sometimes outperform) the training loss trajectory and performance of full-finetuning with notable memory and compute footprint reduction. For example, finetuning a 50\% compressed LLaMa-2 7B model using only a fraction of parameters in LRCs (WeLore) can outperform its full finetuning with ~3x better throughput and ~0.6x GPU requirement. Our codes are available at https://github.com/VITA-Group/welore

With the fast growth of parameter size, it becomes increasingly challenging to deploy large generative models as they typically require large GPU memory consumption and massive computation. Unstructured model pruning has been a common approach to reduce both GPU memory footprint and the overall computation while retaining good model accuracy. However, the existing solutions do not provide a highly-efficient support for handling unstructured sparsity on modern GPUs, especially on the highly-structured Tensor Core hardware. Therefore, we propose Flash-LLM for enabling low-cost and highly-efficient large generative model inference with the sophisticated support of unstructured sparsity on high-performance but highly restrictive Tensor Cores. Based on our key observation that the main bottleneck of generative model inference is the several skinny matrix multiplications for which Tensor Cores would be significantly under-utilized due to low computational intensity, we propose a general Load-as-Sparse and Compute-as-Dense methodology for unstructured sparse matrix multiplication. The basic insight is to address the significant memory bandwidth bottleneck while tolerating redundant computations that are not critical for end-to-end performance on Tensor Cores. Based on this, we design an effective software framework for Tensor Core based unstructured SpMM, leveraging on-chip resources for efficient sparse data extraction and computation/memory-access overlapping. At SpMM kernel level, Flash-LLM significantly outperforms the state-of-the-art library, i.e., Sputnik and SparTA by an average of 2.9x and 1.5x, respectively. At end-to-end framework level on OPT-30B/66B/175B models, for tokens per GPU-second, Flash-LLM achieves up to 3.8x and 3.6x improvement over DeepSpeed and FasterTransformer, respectively, with significantly lower inference cost.

Large language models (LLMs) have shown impressive performance on language tasks but face challenges when deployed on resource-constrained devices due to their extensive parameters and reliance on dense multiplications, resulting in high memory demands and latency bottlenecks. Shift-and-add reparameterization offers a promising solution by replacing costly multiplications with hardware-friendly primitives in both the attention and multi-layer perceptron (MLP) layers of an LLM. However, current reparameterization techniques require training from scratch or full parameter fine-tuning to restore accuracy, which is resource-intensive for LLMs. To address this, we propose accelerating pretrained LLMs through post-training shift-and-add reparameterization, creating efficient multiplication-free models, dubbed ShiftAddLLM. Specifically, we quantize each weight matrix into binary matrices paired with group-wise scaling factors. The associated multiplications are reparameterized into (1) shifts between activations and scaling factors and (2) queries and adds according to the binary matrices. To reduce accuracy loss, we present a multi-objective optimization method to minimize both weight and output activation reparameterization errors. Additionally, based on varying sensitivity across layers to reparameterization, we develop an automated bit allocation strategy to further reduce memory usage and latency. Experiments on five LLM families and eight tasks consistently validate the effectiveness of ShiftAddLLM, achieving average perplexity improvements of 5.6 and 22.7 points at comparable or lower latency compared to the most competitive quantized LLMs at 3 and 2 bits, respectively, and more than 80% memory and energy reductions over the original LLMs. Codes and models are available at https://github.com/GATECH-EIC/ShiftAddLLM.

NVFP4 is an attractive datatype for large language model (LLM) inference, offering compact storage and native tensor-core acceleration. However, preserving accuracy using NVFP4 requires careful quantization. In this work we develop a novel quantization scheme called Optimized Smoothing and Scaling for NVFP4 (OSFP4). For each linear projection it uses a diagonal smoothing matrix whose entries are optimized to minimize the squared matrix-product quantization error under NVFP4, taking into account the rounding procedure that is used (either round-to-nearest, or GPTQ-style successive interference cancellation). This requires performing joint optimization on the smoothing entries as well as the block scales, which is facilitated by analyzing a multiplicative-dither FP4 quantizer instead of the fixed deterministic one. Experiments show that OSFP4 achieves the highest average accuracy among the evaluated competitors in the corresponding quantization settings, while retaining approximately 94-97\% of vendor NVFP4 prefill throughput on the measured workloads. Our code is available in https://github.com/neriahbd/OSFP4

Large language model (LLM) decoding is latency-sensitive and often bottlenecked by fragmented operator execution and repeated off-chip materialization of intermediate tensors. Prior work expands fusion scope by leveraging thread-block clusters and on-chip inter-block collectives to fuse attention-side operators such as QKV projection, attention, and output projection. We develop ClusterFusion++, a CUDA-level extension that broadens fusion to the full Transformer decoder block for GPT-NeoX/Pythia models: LayerNorm -> QKV -> RoPE -> decode attention -> output projection -> Post-LN -> MLP -> residual. We additionally engineer a CUDA-Graph-compatible execution mode with persistent Tensor Memory Accelerator (TMA) descriptors to reduce per-step overhead. On an NVIDIA RTX 5090-class GPU, ClusterFusion++ improves throughput by 1.34x for Pythia-2.8B and yields similar gains for Pythia-6.9B, while maintaining high output fidelity (near-token-identical generation, with minor non-determinism from FP16 atomics).

Dynamic Sparse Training (DST) methods achieve state-of-the-art results in sparse neural network training, matching the generalization of dense models while enabling sparse training and inference. Although the resulting models are highly sparse and theoretically less computationally expensive, achieving speedups with unstructured sparsity on real-world hardware is challenging. In this work, we propose a sparse-to-sparse DST method, Structured RigL (SRigL), to learn a variant of fine-grained structured N:M sparsity by imposing a constant fan-in constraint. Using our empirical analysis of existing DST methods at high sparsity, we additionally employ a neuron ablation method which enables SRigL to achieve state-of-the-art sparse-to-sparse structured DST performance on a variety of Neural Network (NN) architectures. We demonstrate reduced real-world timings on CPU for online inference -- 3.6x/2x faster at 90% sparsity than equivalent dense/unstructured sparse layers, respectively. Our source code is available at https://github.com/calgaryml/condensed-sparsity

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions typically either rescale updates after orthogonalization or use heavier whitening-based preconditioners before it. We introduce {\method}, a lightweight family of pre-orthogonalization equilibration schemes for Muon with three forms: two-sided row/column normalization (RC), row normalization (R), and column normalization (C). By rebalancing the momentum matrix before finite-step Newton--Schulz orthogonalization, {\method} improves the geometry seen by orthogonalization. We show that finite-step orthogonalization is governed by the input spectrum, especially stable rank and condition number, and that row/column normalization acts as a zeroth-order surrogate for whitening. For hidden matrix weights, R is the default variant. Theoretically, {\method} (R) retains the standard mathcal O(T^{-1/4}) Muon-type nonconvex stationarity guarantee with decoupled weight decay and a horizon-free diminishing learning-rate schedule, and extends it to finite-step NS5 up to an explicit inexactness constant. In LLaMA2 pretraining on C4, {\method} (R) consistently outperforms Muon on 130M, 350M, and 1B models, with faster convergence and lower validation perplexity. The code is available at the https://github.com/MaeChd/muon-eq{MuonEq codebase}.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

Knowledge graphs are incomplete by nature, with only a limited number of observed facts from the world knowledge being represented as structured relations between entities. To partly address this issue, an important task in statistical relational learning is that of link prediction or knowledge graph completion. Both linear and non-linear models have been proposed to solve the problem. Bilinear models, while expressive, are prone to overfitting and lead to quadratic growth of parameters in number of relations. Simpler models have become more standard, with certain constraints on bilinear map as relation parameters. In this work, we propose a factorized bilinear pooling model, commonly used in multi-modal learning, for better fusion of entities and relations, leading to an efficient and constraint-free model. We prove that our model is fully expressive, providing bounds on the embedding dimensionality and factorization rank. Our model naturally generalizes Tucker decomposition based TuckER model, which has been shown to generalize other models, as efficient low-rank approximation without substantially compromising the performance. Due to low-rank approximation, the model complexity can be controlled by the factorization rank, avoiding the possible cubic growth of TuckER. Empirically, we evaluate on real-world datasets, reaching on par or state-of-the-art performance. At extreme low-ranks, model preserves the performance while staying parameter efficient.

In a compound AI system, components such as an LLM call, a retriever, a code interpreter, or tools are interconnected. The system's behavior is primarily driven by parameters such as instructions or tool definitions. Recent advancements enable end-to-end optimization of these parameters using an LLM. Notably, leveraging an LLM as an optimizer is particularly efficient because it avoids gradient computation and can generate complex code and instructions. This paper presents a survey of the principles and emerging trends in LLM-based optimization of compound AI systems. It covers archetypes of compound AI systems, approaches to LLM-based end-to-end optimization, and insights into future directions and broader impacts. Importantly, this survey uses concepts from program analysis to provide a unified view of how an LLM optimizer is prompted to optimize a compound AI system. The exhaustive list of paper is provided at https://github.com/linyuhongg/LLM-based-Optimization-of-Compound-AI-Systems.

Large Language Models (LLMs) have achieved human-level proficiency across diverse tasks, but their ability to perform rigorous mathematical problem solving remains an open challenge. In this work, we investigate a fundamental yet computationally intractable problem: determining whether a given multivariate polynomial is nonnegative. This problem, closely related to Hilbert's Seventeenth Problem, plays a crucial role in global polynomial optimization and has applications in various fields. First, we introduce SoS-1K, a meticulously curated dataset of approximately 1,000 polynomials, along with expert-designed reasoning instructions based on five progressively challenging criteria. Evaluating multiple state-of-the-art LLMs, we find that without structured guidance, all models perform only slightly above the random guess baseline 50%. However, high-quality reasoning instructions significantly improve accuracy, boosting performance up to 81%. Furthermore, our 7B model, SoS-7B, fine-tuned on SoS-1K for just 4 hours, outperforms the 671B DeepSeek-V3 and GPT-4o-mini in accuracy while only requiring 1.8% and 5% of the computation time needed for letters, respectively. Our findings highlight the potential of LLMs to push the boundaries of mathematical reasoning and tackle NP-hard problems.