Daily Papers

The advancements in Large Language Models (LLMs) have been hindered by their substantial sizes, which necessitate LLM compression methods for practical deployment. Singular Value Decomposition (SVD) offers a promising solution for LLM compression. However, state-of-the-art SVD-based LLM compression methods have two key limitations: truncating smaller singular values may lead to higher compression loss, and the lack of update on the remaining model parameters after SVD truncation. In this work, we propose SVD-LLM, a new SVD-based LLM compression method that addresses the limitations of existing methods. SVD-LLM incorporates a truncation-aware data whitening strategy to ensure a direct mapping between singular values and compression loss. Moreover, SVD-LLM adopts a layer-wise closed-form model parameter update strategy to compensate for accuracy degradation caused by SVD truncation. We evaluate SVD-LLM on a total of 11 datasets and seven models from three different LLM families at four different scales. Our results demonstrate the superiority of SVD-LLM over state-of-the-arts, especially at high model compression ratios. The source code is available at https://github.com/AIoT-MLSys-Lab/SVD-LLM.

Large pre-trained Transformer models achieve state-of-the-art results across diverse language and reasoning tasks, but full fine-tuning incurs substantial storage, memory, and computational overhead. Parameter-efficient fine-tuning (PEFT) methods mitigate these costs by learning only a small subset of task-specific parameters, yet existing approaches either introduce inference-time latency (adapter modules), suffer from suboptimal convergence (randomly initialized low-rank updates), or rely on fixed rank choices that may not match task complexity (Kronecker-based decompositions). We propose SoKA (SVD on Kronecker Adaptation), a novel PEFT strategy that combines Kronecker-product tensor factorization with SVD-driven initialization and spectrum-aware dynamic rank selection. Our Kronecker-Product SVD (KPSVD) procedure extracts principal components of the full weight update into compact Kronecker factors, while an adaptive rank selection algorithm uses energy-threshold and elbow-point criteria to prune negligible components. Empirical evaluation on LLaMA2-7B across arithmetic reasoning (GSM8K), formal mathematics (MATH), and code generation (MBPP) demonstrates that SoKA requires only 0.99M trainable parameters, 25% fewer than LoRA/PiSSA, while matching or exceeding baseline performance. Moreover, SoKA exhibits faster convergence and more stable gradients, highlighting its robustness and efficiency for large-scale model adaptation.

Singular Value Decomposition (SVD) constitutes a bridge between the linear algebra concepts and multi-layer neural networks---it is their linear analogy. Besides of this insight, it can be used as a good initial guess for the network parameters, leading to substantially better optimization results.

Vision-language models (VLMs) like CLIP have shown impressive zero-shot and few-shot learning capabilities across diverse applications. However, adapting these models to new fine-grained domains remains difficult due to reliance on prompt engineering and the high cost of full model fine-tuning. Existing adaptation approaches rely on augmented components, such as prompt tokens and adapter modules, which could limit adaptation quality, destabilize the model, and compromise the rich knowledge learned during pretraining. In this work, we present CLIP-SVD, a novel multi-modal and parameter-efficient adaptation technique that leverages Singular Value Decomposition (SVD) to modify the internal parameter space of CLIP without injecting additional modules. Specifically, we fine-tune only the singular values of the CLIP parameter matrices to rescale the basis vectors for domain adaptation while retaining the pretrained model. This design enables enhanced adaptation performance using only 0.04\% of the model's total parameters and better preservation of its generalization ability. CLIP-SVD achieves state-of-the-art classification results on 11 natural and 10 biomedical datasets, outperforming previous methods in both accuracy and generalization under few-shot settings. Additionally, we leverage a natural language-based approach to analyze the effectiveness and dynamics of the CLIP adaptation to allow interpretability of CLIP-SVD. The code is publicly available at https://github.com/HealthX-Lab/CLIP-SVD.

Despite significant advancements, the practical deployment of Large Language Models (LLMs) is often hampered by their immense sizes, highlighting the need for effective compression techniques. Singular Value Decomposition (SVD) is a promising LLM compression technique. However, existing SVD-based compression methods fall short in reducing truncation losses, leading to less competitive performance in compressed models. In this work, we introduce SVD-LLM V2, a SVD-based LLM compression method that optimizes singular value truncation in SVD compression with two techniques. First, SVD-LLM V2 proposes to use theoretical truncation loss of weight matrices to assign a unique compression ratio to each weight matrix at different layers to accommodate weight redundancy heterogeneity. Second, SVD-LLM V2 proposes loss-optimized weight truncation to ensure that the truncated singular values result in a lower and more stable truncation loss in practice. We evaluate SVD-LLM V2 on ten datasets and five LLMs at various scales. Our results show SVD-LLM V2 outperforms state-of-the-art SVD-based LLM compression methods. Our code is available at https://github.com/AIoT-MLSys-Lab/SVD-LLM

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.

Foundation models have become a dominant paradigm in machine learning, achieving remarkable performance across diverse tasks through large-scale pretraining. However, they often yield overconfident, uncalibrated predictions. The standard approach to quantifying epistemic uncertainty are ensembles of multiple independently trained models. But their computational cost scales linearly with ensemble size, making them impractical for large foundation models. We propose Singular Value Ensemble (SVE), a parameter-efficient implicit ensembling method. SVE builds on a simple, but powerful core assumption: namely, that the singular vectors of the weight matrices correspond to meaningful directions in the representation space. If the singular vectors are indeed meaningful (orthogonal) "knowledge directions", then a model ensemble can be obtained by modulating only how strongly each direction contributes to the output. Rather than learning new parameters for each ensemble member, we freeze the singular vectors and only train per-member singular values that rescale the contribution of each direction in that shared knowledge basis. Ensemble diversity emerges naturally during joint training as stochastic initialization and random batch sampling cause different members to converge to different combinations of the same underlying knowledge. SVE performs comparable to an explicit ensemble, while increasing the parameter count of the base model by <1%, making principled uncertainty estimation accessible in resource-constrained settings. We validate SVE on NLP and vision tasks with various different backbones and show that it improves calibration while maintaining predictive accuracy.

Representation learning and exploration are among the key challenges for any deep reinforcement learning agent. In this work, we provide a singular value decomposition based method that can be used to obtain representations that preserve the underlying transition structure in the domain. Perhaps interestingly, we show that these representations also capture the relative frequency of state visitations, thereby providing an estimate for pseudo-counts for free. To scale this decomposition method to large-scale domains, we provide an algorithm that never requires building the transition matrix, can make use of deep networks, and also permits mini-batch training. Further, we draw inspiration from predictive state representations and extend our decomposition method to partially observable environments. With experiments on multi-task settings with partially observable domains, we show that the proposed method can not only learn useful representation on DM-Lab-30 environments (that have inputs involving language instructions, pixel images, and rewards, among others) but it can also be effective at hard exploration tasks in DM-Hard-8 environments.

The rapid advancement in large language models (LLMs) comes with a significant increase in their parameter size, presenting challenges for adaptation and fine-tuning. Parameter-efficient fine-tuning (PEFT) methods are widely used to adapt LLMs for downstream tasks efficiently. In this paper, we propose Singular Values and Orthonormal Regularized Singular Vectors Adaptation, or SORSA, a novel PEFT method. We introduce a method to analyze the variation of the parameters by performing singular value decomposition (SVD) and discuss and analyze SORSA's superiority in minimizing the alteration in the SVD aspect. Each SORSA adapter consists of two main parts: trainable principal singular weights W_p = U_p Sigma_p V^top_p, and frozen residual weights W_r = U_r Sigma_r V^top_r. These parts are initialized by performing SVD on pre-trained weights. Moreover, we implement and analyze an orthonormal regularizer, which could effectively transfer the scaling information into Sigma_p and ultimately allows the training process to be more efficient. SORSA adapters could be merged during inference, thus eliminating any inference latency. After all, SORSA shows a faster convergence than PiSSA and LoRA in our experiments. On the MATH benchmark, Llama 2 7B adapted using SORSA achieved 10.36% accuracy, outperforming LoRA (5.50%), Full FT (7.22%), and PiSSA (7.44%). On the GSM-8K benchmark, SORSA achieved 56.03% accuracy, surpassing LoRA (42.30%), Full FT (49.05%), and PiSSA (53.07%). We conclude that SORSA offers a new perspective on parameter-efficient fine-tuning, demonstrating remarkable performance. The code is available at https://github.com/Gunale0926/SORSA.

The complex nature of medical image segmentation calls for models that are specifically designed to capture detailed, domain-specific features. Large foundation models offer considerable flexibility, yet the cost of fine-tuning these models remains a significant barrier. Parameter-Efficient Fine-Tuning (PEFT) methods, such as Low-Rank Adaptation (LoRA), efficiently update model weights with low-rank matrices but may suffer from underfitting when the chosen rank is insufficient to capture domain-specific nuances. Conversely, full-rank Singular Value Decomposition (SVD) based methods provide comprehensive updates by modifying all singular values, yet they often lack flexibility and exhibit variable performance across datasets. We propose SALT (Singular Value Adaptation with Low-Rank Transformation), a method that selectively adapts the most influential singular values using trainable scale and shift parameters while complementing this with a low-rank update for the remaining subspace. This hybrid approach harnesses the advantages of both LoRA and SVD, enabling effective adaptation without relying on increasing model size or depth. Evaluated on 5 challenging medical datasets, ranging from as few as 20 samples to 1000, SALT outperforms state-of-the-art PEFT (LoRA and SVD) by 2% to 5% in Dice with only 3.9% trainable parameters, demonstrating robust adaptation even in low-resource settings. The code for SALT is available at: https://github.com/BioMedIA-MBZUAI/SALT

Domain generalization (DG) aims to adapt a model using one or multiple source domains to ensure robust performance in unseen target domains. Recently, Parameter-Efficient Fine-Tuning (PEFT) of foundation models has shown promising results in the context of DG problem. Nevertheless, existing PEFT methods still struggle to strike a balance between preserving generalizable components of the pre-trained model and learning task-specific features. To gain insights into the distribution of generalizable components, we begin by analyzing the pre-trained weights through the lens of singular value decomposition. Building on these insights, we introduce Singular Value Decomposed Minor Components Adaptation (SoMA), an approach that selectively tunes minor singular components while keeping the residual parts frozen. SoMA effectively retains the generalization ability of the pre-trained model while efficiently acquiring task-specific skills. Moreover, we freeze domain-generalizable blocks and employ an annealing weight decay strategy, thereby achieving an optimal balance in the delicate trade-off between generalizability and discriminability. SoMA attains state-of-the-art results on multiple benchmarks that span both domain generalized semantic segmentation to domain generalized object detection. In addition, our methods introduce no additional inference overhead or regularization loss, maintain compatibility with any backbone or head, and are designed to be versatile, allowing easy integration into a wide range of tasks.

We introduce Sven (Singular Value dEsceNt), a new optimization algorithm for neural networks that exploits the natural decomposition of loss functions into a sum over individual data points, rather than reducing the full loss to a single scalar before computing a parameter update. Sven treats each data point's residual as a separate condition to be satisfied simultaneously, using the Moore-Penrose pseudoinverse of the loss Jacobian to find the minimum-norm parameter update that best satisfies all conditions at once. In practice, this pseudoinverse is approximated via a truncated singular value decomposition, retaining only the k most significant directions and incurring a computational overhead of only a factor of k relative to stochastic gradient descent. This is in comparison to traditional natural gradient methods, which scale as the square of the number of parameters. We show that Sven can be understood as a natural gradient method generalized to the over-parametrized regime, recovering natural gradient descent in the under-parametrized limit. On regression tasks, Sven significantly outperforms standard first-order methods including Adam, converging faster and to a lower final loss, while remaining competitive with LBFGS at a fraction of the wall-time cost. We discuss the primary challenge to scaling, namely memory overhead, and propose mitigation strategies. Beyond standard machine learning benchmarks, we anticipate that Sven will find natural application in scientific computing settings where custom loss functions decompose into several conditions.

The singular value decomposition (SVD) is a powerful tool in modern numerical linear algebra, which underpins computational methods such as principal component analysis (PCA), low-rank approximations, and randomized algorithms. Many practical scenarios require solving numerous small SVD problems, a regime generally referred to as "batch SVD". Existing programming models can handle this efficiently on parallel CPU architectures, but high-performance solutions for GPUs remain immature. A GPU-oriented batch SVD solver is introduced. This solver exploits the one-sided Jacobi algorithm to exploit fine-grained parallelism, and a number of algorithmic and design optimizations achieve unmatched performance. Starting from a baseline solver, a sequence of optimizations is applied to obtain incremental performance gains. Numerical experiments show that the new solver is robust across problems with different numerical properties, matrix shapes, and arithmetic precisions. Performance benchmarks on both NVIDIA and AMD systems show significant performance speedups over vendor solutions as well as existing open-source solvers.

The increasing sizes of large language models (LLMs) result in significant computational overhead and memory usage when adapting these models to specific tasks or domains. Various parameter-efficient fine-tuning (PEFT) methods have been devised to mitigate these challenges by training a small set of parameters for the task-specific updates of the model weights. Among PEFT methods, LoRA stands out for its simplicity and efficiency, inspiring the development of a series of variants. However, LoRA and its successors disregard the knowledge that is noisy or irrelevant to the targeted task, detrimentally impacting model performance and leading to suboptimality. To address this limitation, we introduce Knowledge-aware Singular-value Adaptation (KaSA), a PEFT method that leverages singular value decomposition (SVD) with knowledge-aware singular values to dynamically activate knowledge based on its relevance to the task at hand. We conduct extensive experiments across a range of LLMs on tasks spanning natural language understanding (NLU), generation (NLG), instruction following, and commonsense reasoning. The experimental results demonstrate that KaSA consistently outperforms FFT and 14 popular PEFT baselines across 16 benchmarks and 4 synthetic datasets, underscoring our method's efficacy and adaptability. The source code of our method is available at https://github.com/juyongjiang/KaSA.

In this paper, we introduce a new post-training compression paradigm for Large Language Models (LLMs) to facilitate their wider adoption. We delve into LLM weight low-rank decomposition, and find that the challenges of this task stem from (1) the distribution variance in the LLM activations and (2) the sensitivity difference among various kinds of layers. To address these issues, we propose a training-free approach called Activation-aware Singular Value Decomposition (ASVD). Specifically, ASVD manages activation outliers by transforming the weight matrix based on the activation distribution. This transformation allows the outliers in the activation matrix to be absorbed into the transformed weight matrix, thereby enhancing decomposition accuracy. Additionally, we propose an efficient iterative calibration process to optimize layer-specific decomposition by addressing the varying sensitivity of different LLM layers. In this way, ASVD can compress a network by 10%-30%. Based on the success of the low-rank decomposition of projection matrices in the self-attention module, we further introduce ASVD to compress the KV cache. By reducing the channel dimension of KV activations, memory requirements for KV cache can be largely reduced. ASVD can further achieve 50% KV cache reductions without performance drop in a training-free manner.

Large language models (LLMs) achieve remarkable performance across a wide range of tasks, but their deployment is constrained by substantial memory and compute requirements. Low-rank compression via singular value decomposition (SVD) is an effective remedy, but existing methods focus on how to factorize and which components to keep. We introduce SVD-Surgeon, a training-free method that brings the Optimal Brain Surgeon (OBS) framework to the singular-value basis. Treating each singular value as a parameter, it computes a closed-form update of the retained singular values that compensates, to second order in the model loss, for those removed by truncation. The same analysis yields a saliency for choosing which values to prune. As it operates directly on the singular-value factorization, SVD-Surgeon can be layered on top of existing SVD compressors. Applied to SVD-LLM, a leading SVD-based method, it improves the perplexity-compression trade-off on the OPT family and LLaMA 2-7B without any retraining.

As the parameters of LLMs expand, the computational cost of fine-tuning the entire model becomes prohibitive. To address this challenge, we introduce a PEFT method, Principal Singular values and Singular vectors Adaptation (PiSSA), which optimizes a significantly reduced parameter space while achieving or surpassing the performance of full-parameter fine-tuning. PiSSA is inspired by Intrinsic SAID, which suggests that pre-trained, over-parametrized models inhabit a space of low intrinsic dimension. Consequently, PiSSA represents a matrix W within the model by the product of two trainable matrices A and B, plus a residual matrix W^{res} for error correction. SVD is employed to factorize W, and the principal singular values and vectors of W are utilized to initialize A and B. The residual singular values and vectors initialize the residual matrix W^{res}, which keeps frozen during fine-tuning. Notably, PiSSA shares the same architecture with LoRA. However, LoRA approximates Delta W through the product of two matrices, A, initialized with Gaussian noise, and B, initialized with zeros, while PiSSA initializes A and B with principal singular values and vectors of the original matrix W. PiSSA can better approximate the outcomes of full-parameter fine-tuning at the beginning by changing the essential parts while freezing the "noisy" parts. In comparison, LoRA freezes the original matrix and updates the "noise". This distinction enables PiSSA to convergence much faster than LoRA and also achieve better performance in the end. Due to the same architecture, PiSSA inherits many of LoRA's advantages, such as parameter efficiency and compatibility with quantization. Leveraging a fast SVD method, the initialization of PiSSA takes only a few seconds, inducing negligible cost of switching LoRA to PiSSA.

Large pre-trained models (LPMs) have demonstrated exceptional performance in diverse natural language processing and computer vision tasks. However, fully fine-tuning these models poses substantial memory challenges, particularly in resource-constrained environments. Parameter-efficient fine-tuning (PEFT) methods, such as LoRA, mitigate this issue by adjusting only a small subset of parameters. Nevertheless, these methods typically employ random initialization for low-rank matrices, which can lead to inefficiencies in gradient descent and diminished generalizability due to suboptimal starting points. To address these limitations, we propose SVFit, a novel PEFT approach that leverages singular value decomposition (SVD) to initialize low-rank matrices using critical singular values as trainable parameters. Specifically, SVFit performs SVD on the pre-trained weight matrix to obtain the best rank-r approximation matrix, emphasizing the most critical singular values that capture over 99% of the matrix's information. These top-r singular values are then used as trainable parameters to scale the fundamental subspaces of the matrix, facilitating rapid domain adaptation. Extensive experiments across various pre-trained models in natural language understanding, text-to-image generation, and image classification tasks reveal that SVFit outperforms LoRA while requiring 16 times fewer trainable parameters.

Zero-shot Neural Architecture Search (NAS) typically optimises the architecture search process by exploiting the network or gradient properties at initialisation through zero-cost proxies. The existing proxies often rely on labelled data, which is usually unavailable in real-world settings. Furthermore, the majority of the current methods focus either on optimising the convergence and generalisation attributes or solely on the expressivity of the network architectures. To address both limitations, we first demonstrate how channel collinearity affects the convergence and generalisation properties of a neural network. Then, by incorporating the convergence, generalisation and expressivity in one approach, we propose a zero-cost proxy that omits the requirement of labelled data for its computation. In particular, we leverage the Singular Value Decomposition (SVD) of the neural network layer features and the extrinsic curvature of the network output to design our proxy. %As a result, the proposed proxy is formulated as the simplified harmonic mean of the logarithms of two key components: the sum of the inverse of the feature condition number and the extrinsic curvature of the network output. Our approach enables accurate prediction of network performance on test data using only a single label-free data sample. Our extensive evaluation includes a total of six experiments, including the Convolutional Neural Network (CNN) search space, i.e. DARTS and the Transformer search space, i.e. AutoFormer. The proposed proxy demonstrates a superior performance on multiple correlation benchmarks, including NAS-Bench-101, NAS-Bench-201, and TransNAS-Bench-101-micro; as well as on the NAS task within the DARTS and the AutoFormer search space, all while being notably efficient. The code is available at https://github.com/rohanasthana/Dextr.

Parameter-efficient fine-tuning methods, such as LoRA, reduces the number of trainable parameters. However, they often suffer from scalability issues and differences between their learning pattern and full fine-tuning. To overcome these limitations, we propose Efficient Weight-Decomposed Low-Rank Adaptation (EDoRA): a novel PEFT method that decomposes pre-trained weights into magnitude and directional components. By freezing low-rank matrices, initializing them by singular value decomposition, and introducing a small trainable matrix between them, EDoRA achieves substantial reduction in trainable parameters while maintaining learning capacity. Experimental results on the GLUE benchmark demonstrate that EDoRA achieves competitive or superior performance compared to state-of-the-art methods, such as LoRA and DoRA, with up to 30x fewer trainable parameters. This makes EDoRA a highly efficient solution for adapting LLMs to diverse tasks under memory-constrained settings. Code is available at https://github.com/Hamid-Nasiri/EDoRA .

Despite the groundbreaking success of diffusion models in generating high-fidelity images, their latent space remains relatively under-explored, even though it holds significant promise for enabling versatile and interpretable image editing capabilities. The complicated denoising trajectory and high dimensionality of the latent space make it extremely challenging to interpret. Existing methods mainly explore the feature space of U-Net in Diffusion Models (DMs) instead of the latent space itself. In contrast, we directly investigate the latent space via Singular Value Decomposition (SVD) and discover three useful properties that can be used to control generation results without the requirements of data collection and maintain identity fidelity generated images. Based on these properties, we propose a novel image editing framework that is capable of learning arbitrary attributes from one pair of latent codes destined by text prompts in Stable Diffusion Models. To validate our approach, extensive experiments are conducted to demonstrate its effectiveness and flexibility in image editing. We will release our codes soon to foster further research and applications in this area.

While Low-Rank Adaptation (LoRA) enables parameter-efficient fine-tuning for Large Language Models (LLMs), its performance often falls short of Full Fine-Tuning (Full FT). Current methods optimize LoRA by initializing with static singular value decomposition (SVD) subsets, leading to suboptimal leveraging of pre-trained knowledge. Another path for improving LoRA is incorporating a Mixture-of-Experts (MoE) architecture. However, weight misalignment and complex gradient dynamics make it challenging to adopt SVD prior to the LoRA MoE architecture. To mitigate these issues, we propose Great LoRA Mixture-of-Expert (GOAT), a framework that (1) adaptively integrates relevant priors using an SVD-structured MoE, and (2) aligns optimization with full fine-tuned MoE by deriving a theoretical scaling factor. We demonstrate that proper scaling, without modifying the architecture or training algorithms, boosts LoRA MoE's efficiency and performance. Experiments across 25 datasets, including natural language understanding, commonsense reasoning, image classification, and natural language generation, demonstrate GOAT's state-of-the-art performance, closing the gap with Full FT.

Diffusion transformer (DiT) models have achieved remarkable success in image generation, thanks for their exceptional generative capabilities and scalability. Nonetheless, the iterative nature of diffusion models (DMs) results in high computation complexity, posing challenges for deployment. Although existing cache-based acceleration methods try to utilize the inherent temporal similarity to skip redundant computations of DiT, the lack of correction may induce potential quality degradation. In this paper, we propose increment-calibrated caching, a training-free method for DiT acceleration, where the calibration parameters are generated from the pre-trained model itself with low-rank approximation. To deal with the possible correction failure arising from outlier activations, we introduce channel-aware Singular Value Decomposition (SVD), which further strengthens the calibration effect. Experimental results show that our method always achieve better performance than existing naive caching methods with a similar computation resource budget. When compared with 35-step DDIM, our method eliminates more than 45% computation and improves IS by 12 at the cost of less than 0.06 FID increase. Code is available at https://github.com/ccccczzy/icc.

Low-Rank Adaptation (LoRA) improves downstream performance by restricting task updates to a low-rank parameter subspace, yet how this limited capacity is allocated within a trained adapter remains unclear. Through a geometric and empirical study across multiple tasks and backbones, we find that trained LoRA updates often exhibit an inefficient spectrum: task effects concentrate in a small subset of singular directions, while many remaining components are neutral or detrimental, motivating post-hoc refinement within the learned subspace. We propose Spectral Surgery, a training-free refinement that decomposes a LoRA update with SVD, estimates per-component sensitivity using gradients on a small calibration set, and reweights singular values under a magnitude constraint while keeping the learned directions fixed. Across Llama-3.1-8B and Qwen3-8B on four benchmarks, Spectral Surgery yields consistent gains (up to +4.4 points on CommonsenseQA and +2.4 pass@1 on HumanEval) by adjusting only approx 1{,}000 scalar coefficients. These results demonstrate that SVD-structured, low-cost parameter editing can serve as a practical route to improving trained LoRA adapters in a purely post-hoc manner.

Task Arithmetic has emerged as a simple yet effective method to merge models without additional training. However, by treating entire networks as flat parameter vectors, it overlooks key structural information and is susceptible to task interference. In this paper, we study task vectors at the layer level, focusing on task layer matrices and their singular value decomposition. In particular, we concentrate on the resulting singular vectors, which we refer to as Task Singular Vectors (TSV). Recognizing that layer task matrices are often low-rank, we propose TSV-Compress (TSV-C), a simple procedure that compresses them to 10% of their original size while retaining 99% of accuracy. We further leverage this low-rank space to define a new measure of task interference based on the interaction of singular vectors from different tasks. Building on these findings, we introduce TSV-Merge (TSV-M), a novel model merging approach that combines compression with interference reduction, significantly outperforming existing methods.

This paper introduces an information-aware quantization framework that adaptively compresses the key-value (KV) cache in large language models (LLMs). Although prior work has underscored the distinct roles of key and value cache during inference, our systematic analysis -- examining singular value distributions, spectral norms, and Frobenius norms -- reveals, for the first time, that key matrices consistently exhibit higher norm values and are more sensitive to quantization than value matrices. Furthermore, our theoretical analysis shows that matrices with higher spectral norms amplify quantization errors more significantly. Motivated by these insights, we propose a mixed-precision quantization strategy, KV-AdaQuant, which allocates more bit-width for keys and fewer for values since key matrices have higher norm values. With the same total KV bit budget, this approach effectively mitigates error propagation across transformer layers while achieving significant memory savings. Our extensive experiments on multiple LLMs (1B--70B) demonstrate that our mixed-precision quantization scheme maintains high model accuracy even under aggressive compression. For instance, using 4-bit for Key and 2-bit for Value achieves an accuracy of 75.2%, whereas reversing the assignment (2-bit for Key and 4-bit for Value) yields only 54.7% accuracy. The code is available at https://tinyurl.com/kv-adaquant

Factorizing a large matrix into small matrices is a popular strategy for model compression. Singular value decomposition (SVD) plays a vital role in this compression strategy, approximating a learned matrix with fewer parameters. However, SVD minimizes the squared error toward reconstructing the original matrix without gauging the importance of the parameters, potentially giving a larger reconstruction error for those who affect the task accuracy more. In other words, the optimization objective of SVD is not aligned with the trained model's task accuracy. We analyze this previously unexplored problem, make observations, and address it by introducing Fisher information to weigh the importance of parameters affecting the model prediction. This idea leads to our method: Fisher-Weighted SVD (FWSVD). Although the factorized matrices from our approach do not result in smaller reconstruction errors, we find that our resulting task accuracy is much closer to the original model's performance. We perform analysis with the transformer-based language models, showing our weighted SVD largely alleviates the mismatched optimization objectives and can maintain model performance with a higher compression rate. Our method can directly compress a task-specific model while achieving better performance than other compact model strategies requiring expensive model pre-training. Moreover, the evaluation of compressing an already compact model shows our method can further reduce 9% to 30% parameters with an insignificant impact on task accuracy.

Singular Value Decomposition (SVD) has recently seen a surge of interest as a simple yet powerful tool for large language models (LLMs) compression, with a growing number of works demonstrating 20-80% parameter reductions at minimal accuracy loss. Previous SVD-based approaches have focused primarily on reducing the memory footprint of model weights, largely overlooking the additional activation memory overhead incurred during inference when applying truncated factors via standard dense CUDA kernels. Our experiments demonstrate that this activation overhead, scaling with sequence length and hidden dimension, prevents current SVD compression techniques from achieving any reduction in peak inference memory, thereby limiting their viability for real-world, on-device deployments. We introduce FlashSVD, a novel, end-to-end rank-aware streaming inference framework specifically designed for SVD-compressed large language models. FlashSVD can be seamlessly integrated with any model that employs SVD-based methods for parameter reduction. By fusing low-rank projection kernels directly into both the self-attention and feed-forward network (FFN) pipelines, FlashSVD avoid materializing full-size activation buffers. Instead, small tiles of the truncated factors are loaded into on-chip SRAM, multiplied and reduced on the fly, and immediately evicted, preserving high GPU occupancy and adding no extra latency. On standard encoder benchmarks (e.g., BERT-Base), FlashSVD cuts peak activation memory by up to 70.2% and intermediate transient memory by 75%, all while incur no accuracy loss with upstreaming compression methods, offering a practical path toward memory-constrained deployment of low-rank LLMs.

Large language models (LLMs) have rapidly scaled in size, bringing severe memory and computational challenges that hinder their deployment. Singular Value Decomposition (SVD)-based compression has emerged as an appealing post-training compression technique for LLMs, yet most existing methods apply a uniform compression ratio across all layers, implicitly assuming homogeneous information included in various layers. This overlooks the substantial intra-layer heterogeneity observed in LLMs, where middle layers tend to encode richer information while early and late layers are more redundant. In this work, we revisit the existing SVD-based compression method and propose D-Rank, a framework with layer-wise balanced Dynamic Rank allocation for LLMs compression. We first introduce effective rank as a principled metric to measure the information density of weight matrices, and then allocate ranks via a Lagrange multiplier-based optimization scheme to adaptively assign more capacity to groups with higher information density under a fixed compression ratio. Moreover, we rebalance the allocated ranks across attention layers to account for their varying importance and extend D-Rank to latest LLMs with grouped-query attention. Extensive experiments on various LLMs with different scales across multiple compression ratios demonstrate that D-Rank consistently outperforms SVD-LLM, ASVD, and Basis Sharing, achieving more than 15 lower perplexity with LLaMA-3-8B model on C4 datasets at 20% compression ratio and up to 5% higher zero-shot reasoning accuracy with LLaMA-7B model at 40% compression ratio while achieving even higher throughput.

Singular Value Decomposition (SVD) has become an important technique for reducing the computational burden of Vision Language Models (VLMs), which play a central role in tasks such as image captioning and visual question answering. Although multiple prior works have proposed efficient SVD variants to enable low-rank operations, we find that in practice it remains difficult to achieve substantial latency reduction during model execution. To address this limitation, we introduce a new computational pattern and apply SVD at a finer granularity, enabling real and measurable improvements in execution latency. Furthermore, recognizing that weight elements differ in their relative importance, we adaptively allocate relative importance to each element during SVD process to better preserve accuracy, then extend this framework with quantization applied to both weights and activations, resulting in a highly efficient VLM. Collectively, we introduce~Weighted SVD (WSVD), which outperforms other approaches by achieving over 1.8times decoding speedup while preserving accuracy. We open source our code at: https://github.com/SAI-Lab-NYU/WSVD{https://github.com/SAI-Lab-NYU/WSVD

Large language models (LLMs) have demonstrated impressive capabilities in a wide range of downstream natural language processing tasks. Nevertheless, their considerable sizes and memory demands hinder practical deployment, underscoring the importance of developing efficient compression strategies. Singular value decomposition (SVD) decomposes a matrix into orthogonal components, enabling efficient low-rank approximation. This is particularly suitable for LLM compression, where weight matrices often exhibit significant redundancy. However, current SVD-based methods neglect the residual matrix from truncation, resulting in significant truncation loss. Additionally, compressing all layers of the model results in severe error propagation. To overcome these limitations, we propose ERC-SVD, a new post-training SVD-based LLM compression method from an error-controlled perspective. Specifically, we leverage the residual matrix generated during the truncation process to reduce truncation loss. Moreover, under a fixed overall compression ratio, we selectively compress the last few layers of the model, which mitigates error propagation and improves compressed model performance. Comprehensive evaluations on diverse LLM families and multiple benchmark datasets indicate that ERC-SVD consistently achieves superior performance over existing counterpart methods, demonstrating its practical effectiveness.

Quantum algorithms offer significant speedups over their classical counterparts for a variety of problems. The strongest arguments for this advantage are borne by algorithms for quantum search, quantum phase estimation, and Hamiltonian simulation, which appear as subroutines for large families of composite quantum algorithms. A number of these quantum algorithms were recently tied together by a novel technique known as the quantum singular value transformation (QSVT), which enables one to perform a polynomial transformation of the singular values of a linear operator embedded in a unitary matrix. In the seminal GSLW'19 paper on QSVT [Gily\'en, Su, Low, and Wiebe, ACM STOC 2019], many algorithms are encompassed, including amplitude amplification, methods for the quantum linear systems problem, and quantum simulation. Here, we provide a pedagogical tutorial through these developments, first illustrating how quantum signal processing may be generalized to the quantum eigenvalue transform, from which QSVT naturally emerges. Paralleling GSLW'19, we then employ QSVT to construct intuitive quantum algorithms for search, phase estimation, and Hamiltonian simulation, and also showcase algorithms for the eigenvalue threshold problem and matrix inversion. This overview illustrates how QSVT is a single framework comprising the three major quantum algorithms, thus suggesting a grand unification of quantum algorithms.

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

A basic algorithmic task in automated video surveillance is to separate background and foreground objects. Camera tampering, noisy videos, low frame rate, etc., pose difficulties in solving the problem. A general approach that classifies the tampered frames, and performs subsequent analysis on the remaining frames after discarding the tampered ones, results in loss of information. Several robust methods based on robust principal component analysis (PCA) have been introduced to solve this problem. To date, considerable effort has been expended to develop robust PCA via Principal Component Pursuit (PCP) methods with reduced computational cost and visually appealing foreground detection. However, the convex optimizations used in these algorithms do not scale well to real-world large datasets due to large matrix inversion steps. Also, an integral component of these foreground detection algorithms is singular value decomposition which is nonrobust. In this paper, we present a new video surveillance background modelling algorithm based on a new robust singular value decomposition technique rSVDdpd which takes care of both these issues. We also demonstrate the superiority of our proposed algorithm on a benchmark dataset and a new real-life video surveillance dataset in the presence of camera tampering. Software codes and additional illustrations are made available at the accompanying website rSVDdpd Homepage (https://subroy13.github.io/rsvddpd-home/)

Graph neural network (GNN) is a powerful learning approach for graph-based recommender systems. Recently, GNNs integrated with contrastive learning have shown superior performance in recommendation with their data augmentation schemes, aiming at dealing with highly sparse data. Despite their success, most existing graph contrastive learning methods either perform stochastic augmentation (e.g., node/edge perturbation) on the user-item interaction graph, or rely on the heuristic-based augmentation techniques (e.g., user clustering) for generating contrastive views. We argue that these methods cannot well preserve the intrinsic semantic structures and are easily biased by the noise perturbation. In this paper, we propose a simple yet effective graph contrastive learning paradigm LightGCL that mitigates these issues impairing the generality and robustness of CL-based recommenders. Our model exclusively utilizes singular value decomposition for contrastive augmentation, which enables the unconstrained structural refinement with global collaborative relation modeling. Experiments conducted on several benchmark datasets demonstrate the significant improvement in performance of our model over the state-of-the-arts. Further analyses demonstrate the superiority of LightGCL's robustness against data sparsity and popularity bias. The source code of our model is available at https://github.com/HKUDS/LightGCL.

Updating a truncated Singular Value Decomposition (SVD) is crucial in representation learning, especially when dealing with large-scale data matrices that continuously evolve in practical scenarios. Aligning SVD-based models with fast-paced updates becomes increasingly important. Existing methods for updating truncated SVDs employ Rayleigh-Ritz projection procedures, where projection matrices are augmented based on original singular vectors. However, these methods suffer from inefficiency due to the densification of the update matrix and the application of the projection to all singular vectors. To address these limitations, we introduce a novel method for dynamically approximating the truncated SVD of a sparse and temporally evolving matrix. Our approach leverages sparsity in the orthogonalization process of augmented matrices and utilizes an extended decomposition to independently store projections in the column space of singular vectors. Numerical experiments demonstrate a remarkable efficiency improvement of an order of magnitude compared to previous methods. Remarkably, this improvement is achieved while maintaining a comparable precision to existing approaches.

Low-rank adaptation (LoRA) is a popular method for fine-tuning large-scale pre-trained models in downstream tasks by learning low-rank incremental matrices. Though LoRA and its variants effectively reduce the number of trainable parameters compared to full fine-tuning methods, they often overfit training data, resulting in sub-optimal generalization on test data. To address this problem, we introduce BiLoRA, an overfitting-alleviating fine-tuning approach based on bi-level optimization (BLO). BiLoRA employs pseudo singular value decomposition to parameterize low-rank incremental matrices and splits the training of pseudo singular vectors and values across two different subsets of training data. This division, embedded within separate levels of the BLO framework, mitigates the risk of overfitting to a single dataset. Tested on ten datasets covering natural language understanding and generation tasks and applied to various well-known large pre-trained models, BiLoRA significantly outperforms LoRA methods and other fine-tuning approaches, with similar amounts of trainable parameters.

We examine the geometry of neural network training using the Jacobian of trained network parameters with respect to their initial values. Our analysis reveals low-dimensional structure in the training process which is dependent on the input data but largely independent of the labels. We find that the singular value spectrum of the Jacobian matrix consists of three distinctive regions: a "chaotic" region of values orders of magnitude greater than one, a large "bulk" region of values extremely close to one, and a "stable" region of values less than one. Along each bulk direction, the left and right singular vectors are nearly identical, indicating that perturbations to the initialization are carried through training almost unchanged. These perturbations have virtually no effect on the network's output in-distribution, yet do have an effect far out-of-distribution. While the Jacobian applies only locally around a single initialization, we find substantial overlap in bulk subspaces for different random seeds. Our code is available at https://github.com/EleutherAI/training-jacobian

Large Language Models (LLMs) excel in diverse applications but suffer inefficiency due to massive scale. While quantization reduces computational costs, existing methods degrade accuracy in medium-sized LLMs (e.g., Llama-3-8B) due to activation outliers. To address this, we propose QUAD (Quantization with Activation Decomposition), a framework leveraging Singular Value Decomposition (SVD) to suppress activation outliers for effective 4-bit quantization. QUAD estimates activation singular vectors offline using calibration data to construct an orthogonal transformation matrix P, shifting outliers to additional dimensions in full precision while quantizing rest components to 4-bit. Additionally, QUAD enables parameter-efficient fine-tuning via adaptable full-precision outlier weights, narrowing the accuracy gap between quantized and full-precision models. Experiments demonstrate that QUAD achieves 94% ~ 96% accuracy under W4A4 quantization and 98% accuracy with W4A4/A8 and parameter-efficient fine-tuning for Llama-3 and Qwen-2.5 models. Our code is available at https://github.com/hyx1999/Quad{repository}.

While large language models (LLMs) have achieved remarkable performance across a wide range of tasks, their massive scale incurs prohibitive computational and memory costs for pre-training from scratch. Recent studies have investigated the use of low-rank parameterization as a means of reducing model size and training cost. In this context, sparsity is often employed as a complementary technique to recover important information lost in low-rank compression by capturing salient features in the residual space. However, existing approaches typically combine low-rank and sparse components in a simplistic or ad hoc manner, often resulting in undesirable performance degradation compared to full-rank training. In this paper, we propose LOw-rank and Sparse pre-Training (LOST) for LLMs, a novel method that ingeniously integrates low-rank and sparse structures to enable effective training of LLMs from scratch under strict efficiency constraints. LOST applies singular value decomposition to weight matrices, preserving the dominant low-rank components, while allocating the remaining singular values to construct channel-wise sparse components to complement the expressiveness of low-rank training. We evaluate LOST on LLM pretraining ranging from 60M to 7B parameters. Our experiments show that LOST achieves competitive or superior performance compared to full-rank models, while significantly reducing both memory and compute overhead. Moreover, Code is available at https://github.com/JiaxiLi1/LOST-Low-rank-and-Sparse-Training-for-Large-Language-Models{LOST Repo}

While transfer learning is an advantageous strategy, it overlooks the opportunity to leverage knowledge from numerous available models online. Addressing this multi-source transfer learning problem is a promising path to boost adaptability and cut re-training costs. However, existing approaches are inherently coarse-grained, lacking the necessary precision for granular knowledge extraction and the aggregation efficiency required to fuse knowledge from either a large number of source models or those with high parameter counts. We address these limitations by leveraging Singular Value Decomposition (SVD) to first decompose each source model into its elementary, rank-one components. A subsequent aggregation stage then selects only the most salient components from all sources, thereby overcoming the previous efficiency and precision limitations. To best preserve and leverage the synthesized knowledge base, our method adapts to the target task by fine-tuning only the principal singular values of the merged matrix. In essence, this process only recalibrates the importance of top SVD components. The proposed framework allows for efficient transfer learning, is robust to perturbations both at the input level and in the parameter space (e.g., noisy or pruned sources), and scales well computationally.

Empirical scaling laws prescribe how to allocate parameters, data, and compute, while maximal-update parameterization (muP) enables learning-rate transfer across widths by equalizing early-time update magnitudes. However, in modern scale-invariant architectures, training quickly enters an optimizer-governed steady state where normalization layers create backward scale sensitivity and the effective learning rate becomes width dependent, degrading muP transfer. We address this by introducing a weight-decay scaling rule for AdamW that preserves sublayer gain across widths. Empirically, the singular-value spectrum of each matrix parameter scales in norm as eta/lambda with an approximately invariant shape; under width scaling d, we observe that the top singular value scales approximately as eta/lambdacdot d^{0.75}. Combining this observation with the muP learning-rate rule eta_2propto d^{-1} for matrix-like parameters implies an empirical weight-decay scaling rule lambda_2propto d that approximately keeps sublayer gains width invariant. Together with vector-like parameters trained at eta_1=Theta_d(1) and lambda_1=0, this yields zero-shot transfer of both learning rate and weight decay from proxy to target widths, removing per-width sweeps. We validate the rule on LLaMA-style Transformers and in a minimal synthetic setting, and we provide a simple diagnostic, matching top singular values, to check sublayer-gain invariance. Our results extend muP beyond the near-init regime by explicitly controlling steady-state scales set by the optimizer, offering a practical recipe for width-robust hyperparameter transfer under AdamW.

Fairness remains a critical concern in healthcare, where unequal access to services and treatment outcomes can adversely affect patient health. While Federated Learning (FL) presents a collaborative and privacy-preserving approach to model training, ensuring fairness is challenging due to heterogeneous data across institutions, and current research primarily addresses non-medical applications. To fill this gap, we establish the first experimental benchmark for fairness in medical FL, evaluating six representative FL methods across diverse demographic attributes and imaging modalities. We introduce FairFedMed, the first medical FL dataset specifically designed to study group fairness (i.e., demographics). It comprises two parts: FairFedMed-Oph, featuring 2D fundus and 3D OCT ophthalmology samples with six demographic attributes; and FairFedMed-Chest, which simulates real cross-institutional FL using subsets of CheXpert and MIMIC-CXR. Together, they support both simulated and real-world FL across diverse medical modalities and demographic groups. Existing FL models often underperform on medical images and overlook fairness across demographic groups. To address this, we propose FairLoRA, a fairness-aware FL framework based on SVD-based low-rank approximation. It customizes singular value matrices per demographic group while sharing singular vectors, ensuring both fairness and efficiency. Experimental results on the FairFedMed dataset demonstrate that FairLoRA not only achieves state-of-the-art performance in medical image classification but also significantly improves fairness across diverse populations. Our code and dataset can be accessible via link: https://wang.hms.harvard.edu/fairfedmed/.

We introduce Post-Optimization Model Edit (POME), a new algorithm that enhances the performance of fine-tuned large language models using only their pretrained and fine-tuned checkpoints, without requiring extra data or further optimization. The core idea is to apply a muon-style projection to ΔW, the difference between the fine-tuned and pretrained weights. This projection uses truncated singular value decomposition (SVD) to equalize the influence of dominant update directions and prune small singular values, which often represent noise. As a simple post-processing step, POME is completely decoupled from the training pipeline. It requires zero modifications and imposes no overhead, making it universally compatible with any optimizer or distributed framework. POME delivers consistent gains, boosting average performance by +2.5\% on GSM8K and +1.0\% on code generation. Its broad applicability -- from 7B foundation models to 72B RLHF-instructed models -- establishes it as a practical, zero-cost enhancement for any fine-tuning pipeline. Code is available at https://github.com/NUS-HPC-AI-Lab/POME.

We introduce a data-driven diagnostic that combines the singular value decomposition (SVD) with an information-theoretic entropy to quantify the phase-space complexity of perturbed distribution functions in gyrokinetic turbulence. Applying this framework to nonlinear flux-tube simulations that solve the time evolution of the ion distribution function represented by Fourier modes with the wavenumber for real space, we define the von Neumann entropy (vNE) to analyze the velocity-space structure. A global survey in the wavenumber space reveals a wavenumber-dependent variation of the vNE in velocity-space structure: the vNE remains low at low wavenumber but increases across k_perpρ_{ti}sim 1. Hermite/Laguerre decompositions revealed that the finite Larmor radius (FLR) phase mixing in the perpendicular (magnetic-moment) direction is active. Simultaneously, the systematic increase in vNE for k_perpρ_{ti} correlates with the broadening of the Hermite spectrum, suggesting enhanced parallel phase mixing (Landau resonance) as the primary mechanism for the observed wave number dependence. These results demonstrate that the SVD-based vNE provides a compact measure of kinetic complexity without assuming a predefined basis and enables a global mapping of its wavenumber dependence of phase-mixing processes in gyrokinetic turbulence.

Exponential growth in the scale of modern foundation models has led to the widespread adoption of Low-Rank Adaptation (LoRA) as a parameter-efficient fine-tuning technique. However, standard LoRA implementations disregard the varying intrinsic dimensionality of model layers and enforce a uniform rank, leading to parameter redundancy. We propose Post-Optimization Adaptive Rank Allocation (PARA), a data-free compression method for LoRA that integrates seamlessly into existing fine-tuning pipelines. PARA leverages Singular Value Decomposition to prune LoRA ranks using a global threshold over singular values across all layers. This results in non-uniform rank allocation based on layer-wise spectral importance. As a post-hoc method, PARA circumvents the training modifications and resulting instabilities that dynamic architectures typically incur. We empirically demonstrate that PARA reduces parameter count by 75-90\% while preserving the predictive performance of the original, uncompressed LoRA across multiple vision and language benchmarks. Code will be published upon acceptance.

The phase evolution of gravitational waves encodes critical information about the orbital dynamics of binary systems. In this work, we test the robustness of parameterized tests against unmodeled deviations from general relativity. We demonstrate that these parameterized tests are flexible and sensitive in detecting generic deviations in the waveform using the Cutler-Vallisneri bias formalism. This universality arises from examining the inherent geometry of the waveform signal and understanding how biases manifest. We show how Bayes factors are governed by the intrinsic geometry of the waveform signal manifold when parameterized tests are used to approximate generic violations of GR. We use the singular value decomposition to propose templates that are orthogonal to parameterized tests, identifying degeneracies and enhancing the detection of potential deviations. More broadly, the geometric framework developed here clarifies -- at a fundamental level -- how subtle waveform effects (including orbital eccentricity, spin precession, waveform systematics, and instrumental glitches) can mimic one another in data, and when they are intrinsically distinguishable.

Federated fine-tuning of large language models (LLMs) with low-rank adaptation (LoRA) offers a communication-efficient and privacy-preserving solution for task-specific adaptation. Naive aggregation of LoRA modules introduces noise due to mathematical incorrectness when averaging the downsampling and upsampling matrices independently. However, existing noise-free aggregation strategies inevitably compromise the structural expressiveness of LoRA, limiting its ability to retain client-specific adaptations by either improperly reconstructing the low-rank structure or excluding partially trainable components. We identify this problem as loss of training momentum, where LoRA updates fail to accumulate effectively across rounds, resulting in slower convergence and suboptimal performance. To address this, we propose FedMomentum, a novel framework that enables structured and momentum-preserving LoRA aggregation via singular value decomposition (SVD). Specifically, after aggregating low-rank updates in a mathematically correct manner, FedMomentum applies SVD to extract the dominant components that capture the main update directions. These components are used to reconstruct the LoRA modules with the same rank, while residual components can be retained and later merged into the backbone to preserve semantic information and ensure robustness. Extensive experiments across multiple tasks demonstrate that FedMomentum consistently outperforms prior state-of-the-art methods in convergence speed and final accuracy.

Modern Transformer design and compression both reduce to allocating capacity under a budget. The standard scalars for these decisions, #Params and #FLOPs, capture size and compute but not architectural structure: two architectures with identical parameter budgets but different depth-width, head, or FFN allocations receive identical scores yet behave differently. We propose Neural Spectral Capacity (NSC), a closed-form scalar grounded in the singular-value spectrum of each weight matrix. Under standard random initialization, the Marchenko-Pastur law renders NSC computable from the architectural specification alone, with no model instantiation, data, or gradients. Its layer-wise additive structure admits NSC-DP, an exact dynamic-programming solver returning the architecture globally maximizing NSC under resource constraints in seconds on a CPU -- a guarantee that black-box search over existing training-free proxies cannot provide. Empirically, NSC outperforms #Params, #FLOPs, and representative training-free proxies in ranking across seven Transformer and CNN families (on FlexiBERT, τ= 0.505 on pairs differing in #Params by less than 10%, where #Params collapses to 0.082); NSC-DP discovers a Transformer-XL architecture on WikiText-103 that beats the human-designed baseline in 2 seconds; and prunes LLaMA-7B to the best 5.7B model across eight commonsense reasoning tasks without any calibration data, about 5900x faster than the strongest training-free proxy baseline.

Diffusion Transformers (DiTs) have emerged as a dominant architecture for high-quality text-to-image generation, yet their scale poses challenges for efficient deployment. While truncated singular value decomposition (SVD) is a principled tool for parameter reduction, evidence from large language models (LLMs) suggests that naive low-rank approximation can cause catastrophic failure. In contrast, we find that truncated SVD in DiTs produces smooth degradation even under substantial global compression, with redundancy distributed across projection matrices throughout the whole network rather than concentrated in a few transformer blocks. Building on these insights, we introduce SVDtrunc, a two-step block-level compression scheme, first allocating ranks across blocks and compressing the least important ones via truncated SVD under a global parameter budget, and then fine-tuning all blocks with modular knowledge distillation and a rectified-flow objective. We apply SVDtrunc to FLUX.dev across compression levels ranging from 40-90% of the original parameter count. Across three benchmarks, GenEval, HPSv2, and DPG, we outperform all competing approaches. Notably, and in contrast to prior work, we retain near-full performance at 68% and remain competitive even at 57% of the original parameter budget. Furthermore, we show that SVDtrunc complements step distillation and achieves strong results even without fine-tuning, positioning it as a practical continuation of efficiency improvements beyond diffusion step reduction for large-scale generative models. Project page: https://vislearn.github.io/SVDtrunc/

Adapting large-scale pre-trained generative models in a parameter-efficient manner is gaining traction. Traditional methods like low rank adaptation achieve parameter efficiency by imposing constraints but may not be optimal for tasks requiring high representation capacity. We propose a novel spectrum-aware adaptation framework for generative models. Our method adjusts both singular values and their basis vectors of pretrained weights. Using the Kronecker product and efficient Stiefel optimizers, we achieve parameter-efficient adaptation of orthogonal matrices. We introduce Spectral Orthogonal Decomposition Adaptation (SODA), which balances computational efficiency and representation capacity. Extensive evaluations on text-to-image diffusion models demonstrate SODA's effectiveness, offering a spectrum-aware alternative to existing fine-tuning methods.

Fine-tuning large language models (LLMs) with high parameter efficiency for downstream tasks has become a new paradigm. Low-Rank Adaptation (LoRA) significantly reduces the number of trainable parameters for fine-tuning. Although it has demonstrated commendable performance, updating parameters within a single scale may not be the optimal choice for complex downstream tasks.In this paper, we extend the LoRA to multiple scales, dubbed as LoRA^2. We first combine orthogonal projection theory to train a set of LoRAs in two mutually orthogonal planes. Then, we improve the importance score algorithm, which reduce parameter sensitivity score calculations by approximately 98.5\%. By pruning singular values with lower importance scores, thereby enhancing adaptability to various downstream tasks. Extensive experiments are conducted on two widely used pre-trained models to validate the effectiveness of LoRA^2. Results show that it significantly reduces the number of trainable parameters to just 0.72\% compared to full fine-tuning, while still delivering highly impressive performance. Even when the parameters are further reduced to 0.17M, it still achieves comparable results to the baseline with 8 times more parameters. Our code is available here: https://anonymous.4open.science/r/LoRA-2-5B4C

Layered viscoelastic interiors control the frequency dependence of planetary tidal dissipation, but repeated layered Love-number calculations are expensive in long-term orbital and spin integrations. Equivalent homogeneous rheologies provide a cheaper representation of the same response, but their effective parameters do not by themselves identify the interior layers they represent. We develop a local interpretation of such parameters by differentiating the layered-to-homogeneous construction about a fixed-geometry baseline, perturbing only selected layer rigidities and viscosities. We use the pole-residue representation of the degree-two Love number to construct a physical-to-effective transfer matrix. We then use its singular values, resolution matrix, and mode-dominance matrix to determine which physical perturbations are visible in the Love-number response and how they appear in effective-rheology coordinates. Applied to a five-layer lunar model, the full degree-two response is sensitive to seven of eight solid-layer rheological parameters. The single near-null direction is the inner-core rigidity, consistent with shielding by the fluid outer core and with the weak dependence of Love numbers on core elastic structure. Over the LLR frequency interval, fewer physical directions are visible. To first order, the elastic spring, isolated dashpot, and two Voigt elements reproduce the nine-pole response over this interval. Finite perturbations confirm the tangent prediction to within a few percent, with the validity radius limited by a nearly degenerate pair of long-timescale modes.

Neural Network designs are quite diverse, from VGG-style to ResNet-style, and from Convolutional Neural Networks to Transformers. Towards the design of efficient accelerators, many works have adopted a dataflow-based, inter-layer pipelined architecture, with a customised hardware towards each layer, achieving ultra high throughput and low latency. The deployment of neural networks to such dataflow architecture accelerators is usually hindered by the available on-chip memory as it is desirable to preload the weights of neural networks on-chip to maximise the system performance. To address this, networks are usually compressed before the deployment through methods such as pruning, quantization and tensor decomposition. In this paper, a framework for mapping CNNs onto FPGAs based on a novel tensor decomposition method called Mixed-TD is proposed. The proposed method applies layer-specific Singular Value Decomposition (SVD) and Canonical Polyadic Decomposition (CPD) in a mixed manner, achieving 1.73x to 10.29x throughput per DSP to state-of-the-art CNNs. Our work is open-sourced: https://github.com/Yu-Zhewen/Mixed-TD

Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad\'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. Both methods yield considerable speed-up compared with the SVD or the Newton-Schulz iteration. Experimental results on the de-correlated batch normalization and second-order vision transformer demonstrate that our methods can also achieve competitive and even slightly better performances. The code is available at https://github.com/KingJamesSong/FastDifferentiableMatSqrt{https://github.com/KingJamesSong/FastDifferentiableMatSqrt}.

Optical coherence tomography (OCT) imaging from different camera devices causes challenging domain shifts and can cause a severe drop in accuracy for machine learning models. In this work, we introduce a minimal noise adaptation method based on a singular value decomposition (SVDNA) to overcome the domain gap between target domains from three different device manufacturers in retinal OCT imaging. Our method utilizes the difference in noise structure to successfully bridge the domain gap between different OCT devices and transfer the style from unlabeled target domain images to source images for which manual annotations are available. We demonstrate how this method, despite its simplicity, compares or even outperforms state-of-the-art unsupervised domain adaptation methods for semantic segmentation on a public OCT dataset. SVDNA can be integrated with just a few lines of code into the augmentation pipeline of any network which is in contrast to many state-of-the-art domain adaptation methods which often need to change the underlying model architecture or train a separate style transfer model. The full code implementation for SVDNA is available at https://github.com/ValentinKoch/SVDNA.

Current parameter-efficient fine-tuning (PEFT) methods build adapters without considering the context of downstream task to learn, or the context of important knowledge to maintain. As a result, there is often a performance gap compared to full-parameter finetuning, and meanwhile the finetuned model suffers from catastrophic forgetting of the pre-trained world knowledge. In this paper, we propose CorDA, a Context-oriented Decomposition Adaptation method that builds learnable adapters from weight decomposition oriented by the context of downstream task or world knowledge. Concretely, we collect a few data samples, and perform singular value decomposition for each linear layer of a pre-trained LLM multiplied by the covariance matrix of the input activation using these samples. By doing so, the context of the representative samples is captured through deciding the factorizing orientation. Our method enables two options, the knowledge-preserved adaptation and the instruction-previewed adaptation. For the former, we use question-answering samples to obtain the covariance matrices, and use the decomposed components with the smallest r singular values to initialize a learnable adapter, with the others frozen such that the world knowledge is better preserved. For the latter, we use the instruction data from the finetuning task, such as math or coding, to orientate the decomposition and train the largest r components that capture the main characteristics of the task to learn. We conduct extensive experiments on Math, Code, and Instruction Following tasks. Our knowledge-preserved adaptation not only achieves better performance than LoRA on finetuning tasks, but also mitigates the forgetting of world knowledge. Our instruction-previewed adaptation is able to further enhance the finetuning performance, surpassing full-parameter finetuning and the state-of-the-art PEFT methods.

Existing AI-generated image (AIGI) detection methods often suffer from limited generalization performance. In this paper, we identify a crucial yet previously overlooked asymmetry phenomenon in AIGI detection: during training, models tend to quickly overfit to specific fake patterns in the training set, while other information is not adequately captured, leading to poor generalization when faced with new fake methods. A key insight is to incorporate the rich semantic knowledge embedded within large-scale vision foundation models (VFMs) to expand the previous discriminative space (based on forgery patterns only), such that the discrimination is decided by both forgery and semantic cues, thereby reducing the overfitting to specific forgery patterns. A straightforward solution is to fully fine-tune VFMs, but it risks distorting the well-learned semantic knowledge, pushing the model back toward overfitting. To this end, we design a novel approach called Effort: Efficient orthogonal modeling for generalizable AIGI detection. Specifically, we employ Singular Value Decomposition (SVD) to construct the orthogonal semantic and forgery subspaces. By freezing the principal components and adapting the residual components (sim0.19M parameters), we preserve the original semantic subspace and use its orthogonal subspace for learning forgeries. Extensive experiments on AIGI detection benchmarks demonstrate the superior effectiveness of our approach.

Previous research has shown that the principal singular vectors of a pre-trained model's weight matrices capture critical knowledge. In contrast, those associated with small singular values may contain noise or less reliable information. As a result, the LoRA-based parameter-efficient fine-tuning (PEFT) approach, which does not constrain the use of the spectral space, may not be effective for tasks that demand high representation capacity. In this study, we enhance existing PEFT techniques by incorporating the spectral information of pre-trained weight matrices into the fine-tuning process. We investigate spectral adaptation strategies with a particular focus on the additive adjustment of top singular vectors. This is accomplished by applying singular value decomposition (SVD) to the pre-trained weight matrices and restricting the fine-tuning within the top spectral space. Extensive speaker verification experiments on VoxCeleb1 and CN-Celeb1 demonstrate enhanced tuning performance with the proposed approach. Code is released at https://github.com/lizhepolyu/SpectralFT.

Mixture of Experts (MoE) LLMs face significant obstacles due to their massive parameter scale, which imposes memory, storage, and deployment challenges. Although recent expert merging methods promise greater efficiency by consolidating multiple experts, they are fundamentally hindered by parameter conflicts arising from expert specialization. In this paper, we present Sub-MoE, a novel MoE compression framework via Subspace Expert Merging. Our key insight is to perform joint Singular Value Decomposition (SVD) on concatenated expert weights, reducing conflicting parameters by extracting shared U-matrices while enabling effective merging of the expert-specific V components. Specifically, Sub-MoE consists of two innovative phases: (1) Adaptive Expert Clustering, which groups functionally coherent experts via K-means clustering based on cosine similarity of expert outputs; and (2) Subspace Expert Merging, which first enforces Experts Union Decomposition to derive the shared U-matrix across experts in the same group, then pursues frequency-based merging for individual V-matrices, and finalizes expert reconstruction using the merged V-matrix. In this way, we align and fuse experts in a shared subspace, and can be extended with intra-expert compression for further inference optimization. Extensive experiments on Mixtral, DeepSeek, and Qwen-1.5|3 MoE LLMs demonstrate that our Sub-MoE significantly outperforms existing expert pruning and merging methods. Notably, our Sub-MoE maintains 96\%|86\% of original performance with 25\%|50\% expert reduction on Mixtral-8x7B in zero-shot benchmarks. Code will be released at https://github.com/lliai/MoERazor.

Personalized image generation allows users to preserve styles or subjects of a provided small set of images for further image generation. With the advancement in large text-to-image models, many techniques have been developed to efficiently fine-tune those models for personalization, such as Low Rank Adaptation (LoRA). However, LoRA-based methods often face the challenge of adjusting the rank parameter to achieve satisfactory results. To address this challenge, AutoComponent-LoRA (AC-LoRA) is proposed, which is able to automatically separate the signal component and noise component of the LoRA matrices for fast and efficient personalized artistic style image generation. This method is based on Singular Value Decomposition (SVD) and dynamic heuristics to update the hyperparameters during training. Superior performance over existing methods in overcoming model underfitting or overfitting problems is demonstrated. The results were validated using FID, CLIP, DINO, and ImageReward, achieving an average of 9% improvement.

In the era of large-scale training, model merging has evolved into a tool for creating multitasking models efficiently. It enables the knowledge of models to be fused, without the need for heavy computation as required in traditional multitask learning. Existing merging methods often assume that entries at identical positions in weight matrices serve the same function, enabling straightforward entry-wise comparison and merging. However, this assumption overlooks the complexity of finetuned neural networks, where neurons may develop distinct feature compositions, making direct entry-wise merging problematic. We present Decom-Renorm-Merge (DRM), a simple yet effective approach that leverages Singular Value Decomposition to decompose and coordinate weight matrices into an aligned joint space, where entry-wise merging becomes possible. We showcase the effectiveness of DRM across various settings ranging from smaller encoder-based such as ViT and DeBERTa, encoder-decoder-based such as T5, and larger decoder-based such as Llama3.1-8B. Our experimental results show that DRM outperforms several state-of-the-art merging techniques across full finetuning and low-rank adaptation settings. Moreover, our analysis reveals renormalization as the crucial component for creating a robust and even joint space for merging, significantly contributing to the method's performance.

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang's breakthrough quantum-inspired algorithm for recommendation systems [STOC'19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén, Su, Low, and Wiebe [STOC'19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ell^2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

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.

Continual machine unlearning aims to remove the influence of data that should no longer be retained, while preserving the usefulness of the model on everything else. This setting becomes especially difficult when deletion requests arrive sequentially, because the model must repeatedly adapt without erasing previously retained knowledge. Low-Rank Adaptation (LoRA) offers an efficient way to implement such updates, but naively combining many sequential LoRA modules leads to parameter collision, causing strong interference between tasks. We propose a static alternative based on Singular Value Decomposition (SVD)-guided orthogonal subspace projection. Our method constrains each new LoRA update during training so that it lies in the orthogonal complement of the subspaces used by earlier unlearning tasks. This preserves task isolation without requiring dynamic routing at deployment. Experiments on CIFAR-100 with ResNet-20 and on MNIST show stable behavior across long sequences of unlearning tasks. After thirty sequential unlearning tasks, state-of-the-art static fusion reduces retained accuracy from 60.39\% to 12.70\%, whereas the proposed in-training constrained optimization maintains baseline performance (sim58.1\%) while preserving strong unlearning efficacy.

Diffusion models have achieved remarkable success in text-to-image generation, enabling the creation of high-quality images from text prompts or other modalities. However, existing methods for customizing these models are limited by handling multiple personalized subjects and the risk of overfitting. Moreover, their large number of parameters is inefficient for model storage. In this paper, we propose a novel approach to address these limitations in existing text-to-image diffusion models for personalization. Our method involves fine-tuning the singular values of the weight matrices, leading to a compact and efficient parameter space that reduces the risk of overfitting and language drifting. We also propose a Cut-Mix-Unmix data-augmentation technique to enhance the quality of multi-subject image generation and a simple text-based image editing framework. Our proposed SVDiff method has a significantly smaller model size compared to existing methods (approximately 2,200 times fewer parameters compared with vanilla DreamBooth), making it more practical for real-world applications.

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

Recently, Ainsworth et al. showed that using weight matching (WM) to minimize the L_2 distance in a permutation search of model parameters effectively identifies permutations that satisfy linear mode connectivity (LMC), in which the loss along a linear path between two independently trained models with different seeds remains nearly constant. This paper provides a theoretical analysis of LMC using WM, which is crucial for understanding stochastic gradient descent's effectiveness and its application in areas like model merging. We first experimentally and theoretically show that permutations found by WM do not significantly reduce the L_2 distance between two models and the occurrence of LMC is not merely due to distance reduction by WM in itself. We then provide theoretical insights showing that permutations can change the directions of the singular vectors, but not the singular values, of the weight matrices in each layer. This finding shows that permutations found by WM mainly align the directions of singular vectors associated with large singular values across models. This alignment brings the singular vectors with large singular values, which determine the model functionality, closer between pre-merged and post-merged models, so that the post-merged model retains functionality similar to the pre-merged models, making it easy to satisfy LMC. Finally, we analyze the difference between WM and straight-through estimator (STE), a dataset-dependent permutation search method, and show that WM outperforms STE, especially when merging three or more models.

Fine-tuning is a crucial process for adapting large language models (LLMs) to diverse applications. In certain scenarios, such as multi-tenant serving, deploying multiple LLMs becomes necessary to meet complex demands. Recent studies suggest decomposing a fine-tuned LLM into a base model and corresponding delta weights, which are then compressed using low-rank or low-bit approaches to reduce costs. In this work, we observe that existing low-rank and low-bit compression methods can significantly harm the model performance for task-specific fine-tuned LLMs (e.g., WizardMath for math problems). Motivated by the long-tail distribution of singular values in the delta weights, we propose a delta quantization approach using mixed-precision. This method employs higher-bit representation for singular vectors corresponding to larger singular values. We evaluate our approach on various fine-tuned LLMs, including math LLMs, code LLMs, chat LLMs, and even VLMs. Experimental results demonstrate that our approach performs comparably to full fine-tuned LLMs, surpassing both low-rank and low-bit baselines by a considerable margin. Additionally, we show that our method is compatible with various backbone LLMs, such as Llama-2, Llama-3, and Mistral, highlighting its generalizability.

Approaches for compressing large-language models using low-rank decomposition have made strides, particularly with the introduction of activation and loss-aware SVD, which improves the trade-off between decomposition rank and downstream task performance. Despite these advancements, a persistent challenge remains--selecting the optimal ranks for each layer to jointly optimise compression rate and downstream task accuracy. Current methods either rely on heuristics that can yield sub-optimal results due to their limited discrete search space or are gradient-based but are not as performant as heuristic approaches without post-compression fine-tuning. To address these issues, we propose Learning to Low-Rank Compress (LLRC), a gradient-based approach which directly learns the weights of masks that select singular values in a fine-tuning-free setting. Using a calibration dataset, we train only the mask weights to select fewer and fewer singular values while minimising the divergence of intermediate activations from the original model. Our approach outperforms competing ranking selection methods that similarly require no post-compression fine-tuning across various compression rates on common-sense reasoning and open-domain question-answering tasks. For instance, with a compression rate of 20% on Llama-2-13B, LLRC outperforms the competitive Sensitivity-based Truncation Rank Searching (STRS) on MMLU, BoolQ, and OpenbookQA by 12%, 3.5%, and 4.4%, respectively. Compared to other compression techniques, our approach consistently outperforms fine-tuning-free variants of SVD-LLM and LLM-Pruner across datasets and compression rates. Our fine-tuning-free approach also performs competitively with the fine-tuning variant of LLM-Pruner.

Soft spectral clipping (SSC) reduces gradient singular values, but its effect depends on the gradient scale and clipping threshold. We introduce CAST (Clipping with Adaptive Scales and Thresholds), a history-adaptive extension of SSC. AdamW's previous second moment sets row and column scales. An exponential moving average of the clipped output's estimated spectral norm sets each matrix's threshold. CAST restores the original coordinates before the AdamW update. We train a mixture-of-experts (MoE) model with 19.08B total parameters for 83.89B tokens. CAST achieves lower training and mean validation losses than nine gradient-processing baselines, with the fewest relative gradient spikes. It reaches the best baseline's final smoothed training loss in 7.2\% fewer steps.

In recent years, multiple notions of algorithmic fairness have arisen. One such notion is individual fairness (IF), which requires that individuals who are similar receive similar treatment. In parallel, matrix estimation (ME) has emerged as a natural paradigm for handling noisy data with missing values. In this work, we connect the two concepts. We show that pre-processing data using ME can improve an algorithm's IF without sacrificing performance. Specifically, we show that using a popular ME method known as singular value thresholding (SVT) to pre-process the data provides a strong IF guarantee under appropriate conditions. We then show that, under analogous conditions, SVT pre-processing also yields estimates that are consistent and approximately minimax optimal. As such, the ME pre-processing step does not, under the stated conditions, increase the prediction error of the base algorithm, i.e., does not impose a fairness-performance trade-off. We verify these results on synthetic and real data.

Post-training quantization of Large Language Models (LLMs) is challenging. In this work, we introduce Low-rank Quantization Error Reduction (LQER), which combines quantization and low-rank approximation to recover the model capability. LQER leverages an activation-induced scale matrix to drive the singular value distribution of quantization error towards a desirable distribution, which enables nearly-lossless W4A8 quantization on various LLMs and downstream tasks without the need for knowledge distillation, grid search, or gradient-base iterative optimization. Unlike existing methods, the computation pattern of LQER eliminates the need for specialized Scatter and Gather processes to collect high-precision weights from irregular memory locations. Our W4A8 LLMs achieve near-lossless performance on six popular downstream tasks, while using 1.36times fewer hardware resources than the leading state-of-the-art method. We will open-source our framework once the paper is accepted.

Although it is known that transformer language models (LMs) pass features from early layers to later layers, it is not well understood how this information is represented and routed by the model. By analyzing particular mechanism LMs use to accomplish this, we find that it is also used to recall items from a list, and show that this mechanism can explain an otherwise arbitrary-seeming sensitivity of the model to the order of items in the prompt. Specifically, we find that models write into low-rank subspaces of the residual stream to represent features which are then read out by specific later layers, forming low-rank communication channels between layers. By decomposing attention head weight matrices with the Singular Value Decomposition (SVD), we find that previously described interactions between heads separated by one or more layers can be predicted via analysis of their weight matrices. We show that it is possible to manipulate the internal model representations as well as edit model weights based on the mechanism we discover in order to significantly improve performance on our synthetic Laundry List task, which requires recall from a list, often improving task accuracy by over 20%. Our analysis reveals a surprisingly intricate interpretable structure learned from language model pretraining, and helps us understand why sophisticated LMs sometimes fail in simple domains, facilitating future analysis of more complex behaviors.

Text-to-image generation models can create high-quality images from input prompts. However, they struggle to support the consistent generation of identity-preserving requirements for storytelling. Existing approaches to this problem typically require extensive training in large datasets or additional modifications to the original model architectures. This limits their applicability across different domains and diverse diffusion model configurations. In this paper, we first observe the inherent capability of language models, coined context consistency, to comprehend identity through context with a single prompt. Drawing inspiration from the inherent context consistency, we propose a novel training-free method for consistent text-to-image (T2I) generation, termed "One-Prompt-One-Story" (1Prompt1Story). Our approach 1Prompt1Story concatenates all prompts into a single input for T2I diffusion models, initially preserving character identities. We then refine the generation process using two novel techniques: Singular-Value Reweighting and Identity-Preserving Cross-Attention, ensuring better alignment with the input description for each frame. In our experiments, we compare our method against various existing consistent T2I generation approaches to demonstrate its effectiveness through quantitative metrics and qualitative assessments. Code is available at https://github.com/byliutao/1Prompt1Story.

Recent low-rank training methods, such as GaLore, have significantly reduced the memory required to optimize large language models (LLMs). However, these methods often suffer from time-consuming low-rank projection estimations. In particular, the singular value decomposition (SVD) in GaLore can consume more than 80\% of the total training time. To address this issue, we propose GaLore+, which uses cross-head low-rank projection to reduce the substantial time consumption in estimating low-rank projections for multi-head attention. In addition, we employ randomized subspace iteration to achieve fast SVD. To further enhance performance, we propose sparsely coded residuals to reduce the errors caused by low-rank approximation on the first- and second-order moments of the optimizers and weight updates. We evaluate GaLore+ on arithmetic reasoning and natural language generation datasets. Our experiments demonstrate that GaLore+ delivers superior performance while achieving approximately 4times fine-tuning speed compared to vanilla GaLore.

Large language models (LLMs) have shown impressive performance across a range of natural language processing tasks. However, their vast number of parameters introduces significant memory challenges during training, particularly when using memory-intensive optimizers like Adam. Existing memory-efficient algorithms often rely on techniques such as singular value decomposition projection or weight freezing. While these approaches help alleviate memory constraints, they generally produce suboptimal results compared to full-rank updates. In this paper, we investigate the memory-efficient method beyond low-rank training, proposing a novel solution called Gradient Wavelet Transform (GWT), which applies wavelet transforms to gradients in order to significantly reduce the memory requirements for maintaining optimizer states. We demonstrate that GWT can be seamlessly integrated with memory-intensive optimizers, enabling efficient training without sacrificing performance. Through extensive experiments on both pre-training and fine-tuning tasks, we show that GWT achieves state-of-the-art performance compared with advanced memory-efficient optimizers and full-rank approaches in terms of both memory usage and training performance.

Multimodal Continual Instruction Tuning (MCIT) enables Multimodal Large Language Models (MLLMs) to meet continuously emerging requirements without expensive retraining. MCIT faces two major obstacles: catastrophic forgetting (where old knowledge is forgotten) and negative forward transfer (where the performance of future tasks is degraded). Although existing methods have greatly alleviated catastrophic forgetting, they still suffer from negative forward transfer. We discover a large discrepancy in different input embeddings by performing singular value decomposition (SVD) on input embeddings. This discrepancy results in the model learning irrelevant information for old and pre-trained tasks, leading to catastrophic forgetting and negative forward transfer. To address these issues, we propose Prompt Tuning with Positive Forward Transfer (Fwd-Prompt), a prompt-based method that projects the prompt gradient to the residual space to minimize interference between tasks and to the pre-trained subspace for reusing pre-trained knowledge. Our experiments demonstrate that Fwd-Prompt achieves state-of-the-art performance while updating fewer parameters and requiring no old samples. Our research illuminates the potential of continuously adapting MLLMs to new tasks under the instruction tuning paradigm and encourages future studies to explore MCIT.

Large Language Models (LLMs) with long context windows enable powerful applications but come at the cost of high memory consumption to store the Key and Value states (KV-Cache). Recent studies attempted to merge KV-cache from multiple layers into shared representations, yet these approaches either require expensive pretraining or rely on assumptions of high per-token cosine similarity across layers which generally does not hold in practice. We find that the dominant singular vectors are remarkably well-aligned across multiple layers of the KV-Cache. Exploiting this insight, we propose xKV, a simple post-training method that applies Singular Value Decomposition (SVD) on the KV-Cache of grouped layers. xKV consolidates the KV-Cache of multiple layers into a shared low-rank subspace, significantly reducing KV-Cache sizes. Through extensive evaluations on the RULER long-context benchmark with widely-used LLMs (e.g., Llama-3.1 and Qwen2.5), xKV achieves up to 6.8x higher compression rates than state-of-the-art inter-layer technique while improving accuracy by 2.7%. Moreover, xKV is compatible with the emerging Multi-Head Latent Attention (MLA) (e.g., DeepSeek-Coder-V2), yielding a notable 3x compression rates on coding tasks without performance degradation. These results highlight xKV's strong capability and versatility in addressing memory bottlenecks for long-context LLM inference. Our code is publicly available at: https://github.com/abdelfattah-lab/xKV.

Continual learning in large language models (LLMs) is prone to catastrophic forgetting, where adapting to new tasks significantly degrades performance on previously learned ones. Existing methods typically rely on low-rank, parameter-efficient updates that limit the model's expressivity and introduce additional parameters per task, leading to scalability issues. To address these limitations, we propose a novel continual full fine-tuning approach leveraging adaptive singular value decomposition (SVD). Our method dynamically identifies task-specific low-rank parameter subspaces and constrains updates to be orthogonal to critical directions associated with prior tasks, thus effectively minimizing interference without additional parameter overhead or storing previous task gradients. We evaluate our approach extensively on standard continual learning benchmarks using both encoder-decoder (T5-Large) and decoder-only (LLaMA-2 7B) models, spanning diverse tasks including classification, generation, and reasoning. Empirically, our method achieves state-of-the-art results, up to 7% higher average accuracy than recent baselines like O-LoRA, and notably maintains the model's general linguistic capabilities, instruction-following accuracy, and safety throughout the continual learning process by reducing forgetting to near-negligible levels. Our adaptive SVD framework effectively balances model plasticity and knowledge retention, providing a practical, theoretically grounded, and computationally scalable solution for continual learning scenarios in large language models.

Recent advancements in learning latent codes derived from high-dimensional shapes have demonstrated impressive outcomes in 3D generative modeling. Traditionally, these approaches employ a trained autoencoder to acquire a continuous implicit representation of source shapes, which can be computationally expensive. This paper introduces a novel framework, spectral-domain diffusion for high-quality shape generation SpoDify, that utilizes singular value decomposition (SVD) for shape encoding. The resulting eigenvectors can be stored for subsequent decoding, while generative modeling is performed on the eigenfeatures. This approach efficiently encodes complex meshes into continuous implicit representations, such as encoding a 15k-vertex mesh to a 512-dimensional latent code without learning. Our method exhibits significant advantages in scenarios with limited samples or GPU resources. In mesh generation tasks, our approach produces high-quality shapes that are comparable to state-of-the-art methods.

Video diffusion models (DMs) have enabled high-quality video synthesis. Yet, their substantial computational and memory demands pose serious challenges to real-world deployment, even on high-end GPUs. As a commonly adopted solution, quantization has proven notable success in reducing cost for image DMs, while its direct application to video DMs remains ineffective. In this paper, we present QVGen, a novel quantization-aware training (QAT) framework tailored for high-performance and inference-efficient video DMs under extremely low-bit quantization (e.g., 4-bit or below). We begin with a theoretical analysis demonstrating that reducing the gradient norm is essential to facilitate convergence for QAT. To this end, we introduce auxiliary modules (Phi) to mitigate large quantization errors, leading to significantly enhanced convergence. To eliminate the inference overhead of Phi, we propose a rank-decay strategy that progressively eliminates Phi. Specifically, we repeatedly employ singular value decomposition (SVD) and a proposed rank-based regularization gamma to identify and decay low-contributing components. This strategy retains performance while zeroing out inference overhead. Extensive experiments across 4 state-of-the-art (SOTA) video DMs, with parameter sizes ranging from 1.3B sim14B, show that QVGen is the first to reach full-precision comparable quality under 4-bit settings. Moreover, it significantly outperforms existing methods. For instance, our 3-bit CogVideoX-2B achieves improvements of +25.28 in Dynamic Degree and +8.43 in Scene Consistency on VBench.

Hyperspectral image (HSI) restoration aims at recovering clean images from degraded observations and plays a vital role in downstream tasks. Existing model-based methods have limitations in accurately modeling the complex image characteristics with handcraft priors, and deep learning-based methods suffer from poor generalization ability. To alleviate these issues, this paper proposes an unsupervised HSI restoration framework with pre-trained diffusion model (HIR-Diff), which restores the clean HSIs from the product of two low-rank components, i.e., the reduced image and the coefficient matrix. Specifically, the reduced image, which has a low spectral dimension, lies in the image field and can be inferred from our improved diffusion model where a new guidance function with total variation (TV) prior is designed to ensure that the reduced image can be well sampled. The coefficient matrix can be effectively pre-estimated based on singular value decomposition (SVD) and rank-revealing QR (RRQR) factorization. Furthermore, a novel exponential noise schedule is proposed to accelerate the restoration process (about 5times acceleration for denoising) with little performance decrease. Extensive experimental results validate the superiority of our method in both performance and speed on a variety of HSI restoration tasks, including HSI denoising, noisy HSI super-resolution, and noisy HSI inpainting. The code is available at https://github.com/LiPang/HIRDiff.

Open-source foundation models have seen rapid adoption and development, enabling powerful general-purpose capabilities across diverse domains. However, fine-tuning large foundation models for domain-specific or personalized tasks remains prohibitively expensive for most users due to the significant memory overhead beyond that of inference. We introduce EMLoC, an Emulator-based Memory-efficient fine-tuning framework with LoRA Correction, which enables model fine-tuning within the same memory budget required for inference. EMLoC constructs a task-specific light-weight emulator using activation-aware singular value decomposition (SVD) on a small downstream calibration set. Fine-tuning then is performed on this lightweight emulator via LoRA. To tackle the misalignment between the original model and the compressed emulator, we propose a novel compensation algorithm to correct the fine-tuned LoRA module, which thus can be merged into the original model for inference. EMLoC supports flexible compression ratios and standard training pipelines, making it adaptable to a wide range of applications. Extensive experiments demonstrate that EMLoC outperforms other baselines across multiple datasets and modalities. Moreover, without quantization, EMLoC enables fine-tuning of a 38B model on a single 24GB consumer GPU-bringing efficient and practical model adaptation to individual users.

Sexual dimorphism is a critical factor in many biological and medical research fields. In biomechanics and bioengineering, understanding sex differences is crucial for studying musculoskeletal conditions such as temporomandibular disorder (TMD). This paper focuses on the association between the craniofacial skeletal morphology and temporomandibular joint (TMJ) related masticatory muscle attachments to discern sex differences. Data were collected from 10 male and 11 female cadaver heads to investigate sex-specific relationships between the skull and muscles. We propose a conditional cross-covariance reduction (CCR) model, designed to examine the dynamic association between two sets of random variables conditioned on a third binary variable (e.g., sex), highlighting the most distinctive sex-related relationships between skull and muscle attachments in the human cadaver data. Under the CCR model, we employ a sparse singular value decomposition algorithm and introduce a sequential permutation for selecting sparsity (SPSS) method to select important variables and to determine the optimal number of selected variables.

Continual learning aims to learn multiple tasks sequentially while preserving prior knowledge, but faces the challenge of catastrophic forgetting when adapting to new tasks. Recently, approaches leveraging pre-trained models have gained increasing popularity in mitigating this issue, due to the strong generalization ability of foundation models. To adjust pre-trained models for new tasks, existing methods usually employ low-rank adaptation, which restricts parameter updates to a fixed low-rank subspace. However, constraining the optimization space inherently compromises the model's learning capacity, resulting in inferior performance. To address this limitation, we propose Continuous Subspace Optimization for Continual Learning (CoSO) to fine-tune the model in a series of subspaces rather than a single one. These sequential subspaces are dynamically determined through the singular value decomposition of the gradients. CoSO updates the model by projecting gradients onto these subspaces, ensuring memory-efficient optimization. To mitigate forgetting, the optimization subspace of each task is constrained to be orthogonal to the historical task subspace. During task learning, CoSO maintains a task-specific component that captures the critical update directions for the current task. Upon completing a task, this component is used to update the historical task subspace, laying the groundwork for subsequent learning. Extensive experiments on multiple datasets demonstrate that CoSO significantly outperforms state-of-the-art methods, especially in challenging scenarios with long task sequences.

In the training of large language models, parameter-efficient techniques such as LoRA optimize memory usage and reduce communication overhead and memory usage during the fine-tuning phase. However, applying such techniques directly during the pre-training phase results in poor performance, primarily because the premature implementation of low-rank training significantly reduces model accuracy. Existing methods like ReLoRA and GaLore have attempted to address this challenge by updating the low-rank subspace. However, they still fall short of achieving the accuracy of full-rank training. Specifically, ReLoRA restricts the frequency of updates to preserve optimizer states consistency, hindering its ability to closely approximate full-rank training behavior. Meanwhile, GaLore relies on Singular Value Decomposition (SVD) to approximate the full-rank space, which introduces accuracy loss during the approximation process. In this paper, we introduce SwitchLoRA, a parameter-efficient training technique that frequently and smoothly replaces the trainable parameters of LoRA adapters with alternative parameters. SwitchLoRA updates the low-rank subspace incrementally, targeting only a few dimensions at a time to minimize the impact on optimizer states. This allows a higher update frequency, thereby enhancing accuracy by enabling the updated parameters to more closely mimic full-rank behavior during the pre-training phase. Our results demonstrate that SwitchLoRA actually surpasses full-rank training, reducing perplexity from 15.23 to 15.01 on the LLaMA 1.3B model, while also cutting communication overhead by 54\% and memory usage by 13\%. Furthermore, after full fine-tuning the SwitchLoRA pre-trained model and the full-rank pre-trained model on the GLUE benchmark, the SwitchLoRA pre-trained model showed an average accuracy gain of about 1\% over the full-rank pre-trained model.

Post-training compression of Transformer models commonly relies on truncated singular value decomposition (SVD). However, enforcing a single shared subspace can degrade accuracy even at moderate compression. Sparse dictionary learning provides a more flexible union-of-subspaces representation, but existing approaches often suffer from iterative dictionary and coefficient updates. We propose COMPOT (Calibration-Optimized Matrix Procrustes Orthogonalization for Transformers), a training-free compression framework that uses a small calibration dataset to estimate a sparse weight factorization. COMPOT employs orthogonal dictionaries that enable closed-form Procrustes updates for the dictionary and analytical single-step sparse coding for the coefficients, eliminating iterative optimization. To handle heterogeneous layer sensitivity under a global compression budget, COMPOT further introduces a one-shot dynamic allocation strategy that adaptively redistributes layer-wise compression rates. Extensive experiments across diverse architectures and tasks show that COMPOT consistently delivers a superior quality-compression trade-off over strong low-rank and sparse baselines, while remaining fully compatible with post-training quantization for extreme compression. Code is available https://github.com/mts-ai/COMPOT{here}.

In the setting of continual learning, a network is trained on a sequence of tasks, and suffers from catastrophic forgetting. To balance plasticity and stability of network in continual learning, in this paper, we propose a novel network training algorithm called Adam-NSCL, which sequentially optimizes network parameters in the null space of previous tasks. We first propose two mathematical conditions respectively for achieving network stability and plasticity in continual learning. Based on them, the network training for sequential tasks can be simply achieved by projecting the candidate parameter update into the approximate null space of all previous tasks in the network training process, where the candidate parameter update can be generated by Adam. The approximate null space can be derived by applying singular value decomposition to the uncentered covariance matrix of all input features of previous tasks for each linear layer. For efficiency, the uncentered covariance matrix can be incrementally computed after learning each task. We also empirically verify the rationality of the approximate null space at each linear layer. We apply our approach to training networks for continual learning on benchmark datasets of CIFAR-100 and TinyImageNet, and the results suggest that the proposed approach outperforms or matches the state-ot-the-art continual learning approaches.

Muon is an increasingly widely used optimizer that replaces a gradient G=USV^top with its polar factor UV^top, thereby flattening the singular spectrum. However, full flattening discards singular-value information that may matter for adaptation. We introduce Muon^p, a Muon-style optimizer that instead uses fractional spectral-power updates US^pV^top for rational pin(0,1), interpolating between Muon and gradient descent. To make it practical, we prove that fractional spectral powers cannot be computed by any fixed univariate polynomial iteration, and furthermore derive low-degree odd bivariate recurrences that approximate US^pV^top using only matrix multiplications, preserving Muon's matrix-multiplication-only structure and compute complexity. We show that Muon^p maximizes the linear improvement in loss under the Schatten q-norm for q=1+1{p}. Empirically, Muon^p is especially effective for finetuning: on billion-scale models, Muon^p improves validation perplexity and downstream task performance. We further analyze when Muon^p is less suitable, through the lens of spectral geometry. Our results reveal important insights on when preserving the singular spectrum can bring significant gains, and introduce a principled way to achieve them.

Lower numerical precision reduces storage and memory traffic but raises the perturbation floor. We study rank compensation: reinvesting saved memory in a larger approximation rank. For matrices, the singular-value error identity yields a directly testable sufficient condition requiring the additional singular component to offset the perturbation from storing the rank-augmented approximation in lower precision. On ten SuiteSparse matrices, all 100 truncation-dominated configurations (50 FP32 and 50 FP16) are certified non-increases and strict accuracy wins, with mean error ratio 0.963 and storage ratios 58.8% and 29.4% relative to the FP64 baseline. FP16 failures occur only in tail-rank stress tests near the perturbation floor. At the largest resident matrix-application batch, compensated FP32 and FP16 achieve geometric-mean A100 speedups of 1.28times and 2.12times; neither accelerates the smallest batch. For Tensor-Train (TT) approximation, we give a conditional a posteriori extension based on the measured truncation gain and rounded-core perturbation. Across three-way and six-way synthetic tests, FP32 and FP16 achieve combined accuracy-memory wins in 10 of 20 and 14 of 20 trials. On public hyperspectral tensors and FROSTT top-active subtensors, the corresponding counts are 44 of 60 and 54 of 60; four FP16 Salinas-A tail-stress cases fail. No certified TT case exceeds the FP64 error beyond numerical tolerance. Reconstruction of six public tensors yields geometric-mean compensated speedups of 1.38times (FP32) and 1.94times (FP16). Timings cover resident downstream kernels, not factorization, transfers, or end-to-end acceleration.

Foundation models remain vulnerable to spurious correlations and ``Clever Hans'' strategies. Explainable machine learning can find and remove such strategies for classifiers without metadata. For foundation models, no such option exists yet. We propose Disentangled Diffusion Autoencoders (DiDAE). DiDAE wraps a frozen foundation model in a conditional diffusion decoder. A counterfactual is one closed-form edit along a direction of a disentangled dictionary, followed by decoding. The dictionary can be supervised (Procrustes) or unsupervised (Singular Value Decomposition, Sparse Autoencoders). No gradients are needed, so DiDAE is up to 2000 times faster than the state of the art. We evaluate on six datasets, two synthetic and four real-world. In a desiderata-driven benchmark on three of them, its counterfactuals are on par with or better than the state of the art, and they repair downstream classifiers through Counterfactual Knowledge Distillation (CFKD), where they beat metadata-based correction. The same machinery can rank a pretrained dictionary against a trained classifier. It returns the few directions the classifier actually reads, each causally verified by a counterfactual that flips the decision, and repairs the classifier along those a teacher marks spurious. The workflow is plug-and-play in our open-source Peal library we publish alongside the paper. With a public dictionary and a pretrained decoder, all that remains is a cheap linear distillation of the classifier and its own fine-tuning.

Unregistered hyperspectral image (HSI) super-resolution (SR) typically aims to enhance a low-resolution HSI using an unregistered high-resolution reference image. In this paper, we propose an unmixing-based fusion framework that decouples spatial-spectral information to simultaneously mitigate the impact of unregistered fusion and enhance the learnability of SR models. Specifically, we first utilize singular value decomposition for initial spectral unmixing, preserving the original endmembers while dedicating the subsequent network to enhancing the initial abundance map. To leverage the spatial texture of the unregistered reference, we introduce a coarse-to-fine deformable aggregation module, which first estimates a pixel-level flow and a similarity map using a coarse pyramid predictor. It further performs fine sub-pixel refinement to achieve deformable aggregation of the reference features. The aggregative features are then refined via a series of spatial-channel abundance cross-attention blocks. Furthermore, a spatial-channel modulated fusion module is presented to merge encoder-decoder features using dynamic gating weights, yielding a high-quality, high-resolution HSI. Experimental results on simulated and real datasets confirm that our proposed method achieves state-of-the-art super-resolution performance. The code will be available at https://github.com/yingkai-zhang/UAFL.

Fine-tuning large pre-trained language models on downstream tasks has become an important paradigm in NLP. However, common practice fine-tunes all of the parameters in a pre-trained model, which becomes prohibitive when a large number of downstream tasks are present. Therefore, many fine-tuning methods are proposed to learn incremental updates of pre-trained weights in a parameter efficient way, e.g., low-rank increments. These methods often evenly distribute the budget of incremental updates across all pre-trained weight matrices, and overlook the varying importance of different weight parameters. As a consequence, the fine-tuning performance is suboptimal. To bridge this gap, we propose AdaLoRA, which adaptively allocates the parameter budget among weight matrices according to their importance score. In particular, AdaLoRA parameterizes the incremental updates in the form of singular value decomposition. Such a novel approach allows us to effectively prune the singular values of unimportant updates, which is essentially to reduce their parameter budget but circumvent intensive exact SVD computations. We conduct extensive experiments with several pre-trained models on natural language processing, question answering, and natural language generation to validate the effectiveness of AdaLoRA. Results demonstrate that AdaLoRA manifests notable improvement over baselines, especially in the low budget settings. Our code is publicly available at https://github.com/QingruZhang/AdaLoRA .

Multimodal representation learning seeks to create a unified representation space by integrating diverse data modalities to improve multimodal understanding. Traditional methods often depend on pairwise contrastive learning, which relies on a predefined anchor modality, restricting alignment across all modalities. Recent advances have investigated the simultaneous alignment of multiple modalities, yet several challenges remain, such as limitations imposed by fixed anchor points and instability arising from optimizing the product of singular values. To address the challenges, in this paper, we propose Principled Multimodal Representation Learning (PMRL), a novel framework that achieves simultaneous alignment of multiple modalities without anchor dependency in a more stable manner. Specifically, grounded in the theoretical insight that full alignment corresponds to a rank-1 Gram matrix, PMRL optimizes the dominant singular value of the representation matrix to align modalities along a shared leading direction. We propose a softmax-based loss function that treats singular values as logits to prioritize the largest singular value. Besides, instance-wise contrastive regularization on the leading eigenvectors maintains inter-instance separability and prevents representation collapse. Extensive experiments across diverse tasks demonstrate PMRL's superiority compared to baseline methods. Source code can be found in https://github.com/Xiaohao-Liu/PMRL.

We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential distributions and indeed even large sub-matrices of side-length o(n^{1/2-ε}) behave like independent exponentials. We determine the limiting empirical distribution of the singular values the the matrix. Finally the mixing time of the associated Markov chains is shown to be exactly 2 with high probability.

With the proliferation of task-specific large language models, delta compression has emerged as a method to mitigate the resource challenges of deploying numerous such models by effectively compressing the delta model parameters. Previous delta-sparsification methods either remove parameters randomly or truncate singular vectors directly after singular value decomposition (SVD). However, these methods either disregard parameter importance entirely or evaluate it with too coarse a granularity. In this work, we introduce ImPart, a novel importance-aware delta sparsification approach. Leveraging SVD, it dynamically adjusts sparsity ratios of different singular vectors based on their importance, effectively retaining crucial task-specific knowledge even at high sparsity ratios. Experiments show that ImPart achieves state-of-the-art delta sparsification performance, demonstrating 2times higher compression ratio than baselines at the same performance level. When integrated with existing methods, ImPart sets a new state-of-the-art on delta quantization and model merging.

We study the sparse recovery problem with an underdetermined linear system characterized by a Kronecker-structured dictionary and a Kronecker-supported sparse vector. We cast this problem into the sparse Bayesian learning (SBL) framework and rely on the expectation-maximization method for a solution. To this end, we model the Kronecker-structured support with a hierarchical Gaussian prior distribution parameterized by a Kronecker-structured hyperparameter, leading to a non-convex optimization problem. The optimization problem is solved using the alternating minimization (AM) method and a singular value decomposition (SVD)-based method, resulting in two algorithms. Further, we analytically guarantee that the AM-based method converges to the stationary point of the SBL cost function. The SVD-based method, though it adopts approximations, is empirically shown to be more efficient and accurate. We then apply our algorithm to estimate the uplink wireless channel in an intelligent reflecting surface-aided MIMO system and extend the AM-based algorithm to address block sparsity in the channel. We also study the SBL cost to show that the minima of the cost function are achieved at sparse solutions and that incorporating the Kronecker structure reduces the number of local minima of the SBL cost function. Our numerical results demonstrate the effectiveness of our algorithms compared to the state-of-the-art.

Loss of plasticity is a phenomenon where neural networks become more difficult to train during the course of learning. Continual learning algorithms seek to mitigate this effect by sustaining good predictive performance while maintaining network trainability. We develop new techniques for improving continual learning by first reconsidering how initialization can ensure trainability during early phases of learning. From this perspective, we derive new regularization strategies for continual learning that ensure beneficial initialization properties are better maintained throughout training. In particular, we investigate two new regularization techniques for continual learning: (i) Wasserstein regularization toward the initial weight distribution, which is less restrictive than regularizing toward initial weights; and (ii) regularizing weight matrix singular values, which directly ensures gradient diversity is maintained throughout training. We present an experimental analysis that shows these alternative regularizers can improve continual learning performance across a range of supervised learning tasks and model architectures. The alternative regularizers prove to be less sensitive to hyperparameters while demonstrating better training in individual tasks, sustaining trainability as new tasks arrive, and achieving better generalization performance.

Training Large Language Models (LLMs) is memory-intensive due to the large number of parameters and associated optimization states. GaLore, a recent method, reduces memory usage by projecting weight gradients into a low-rank subspace without compromising performance. However, GaLore relies on time-consuming Singular Value Decomposition (SVD) operations to identify the subspace, and the frequent subspace updates lead to significant training time overhead. Moreover, GaLore offers minimal improvements in accuracy and efficiency compared to LoRA in more accessible fine-tuning scenarios. To address these limitations, we introduce Q-Galore, a novel approach that substantially reduces memory usage by combining quantization and low-rank projection, surpassing the benefits of GaLore. Our method is based on two key observations: (i) the gradient subspace exhibits diverse properties, with some layers converging early in training while others are subject to frequent changes; (ii) the projection matrices are highly resilient to low-bit quantization. Leveraging these insights, Q-GaLore adaptively updates the gradient subspace based on its convergence statistics, achieving comparable performance while significantly reducing the number of SVD operations. We maintain the projection matrices in INT4 format and weights in INT8 format, incorporating stochastic rounding to capture accumulated gradient information. This approach enables a high-precision training trajectory using only low-precision weights. We demonstrate that Q-GaLore achieves highly competitive performance with exceptional memory efficiency. At pre-training, Q-GaLore facilitates training a LLaMA-7B model from scratch on a single NVIDIA RTX 4060 Ti with only 16 GB memory. At fine-tuning, it reduces memory consumption by up to 50% compared to LoRA and GaLore, while consistently outperforming QLoRA at the same memory cost.

By leveraging the power of Large Language Models(LLMs) and speech foundation models, state of the art speech-text bimodal works can achieve challenging tasks like spoken translation(ST) and question answering(SQA) altogether with much simpler architectures. In this paper, we utilize the capability of Whisper encoder and pre-trained Yi-6B. Empirical results reveal that modal alignment can be achieved with one layer module and hundred hours of speech-text multitask corpus. We further swap the Yi-6B with human preferences aligned version of Yi-6B-Chat during inference, and discover that the alignment capability is applicable as well. In addition, the alignment subspace revealed by singular value decomposition(SVD) also implies linear alignment subspace is sparse, which leaves the possibility to concatenate other features like voice-print or video to expand modality.

Fine-tuning is a crucial paradigm for adapting pre-trained large language models to downstream tasks. Recently, methods like Low-Rank Adaptation (LoRA) have been shown to match the performance of fully fine-tuned models on various tasks with an extreme reduction in the number of trainable parameters. Even in settings where both methods learn similarly accurate models, are their learned solutions really equivalent? We study how different fine-tuning methods change pre-trained models by analyzing the model's weight matrices through the lens of their spectral properties. We find that full fine-tuning and LoRA yield weight matrices whose singular value decompositions exhibit very different structure; moreover, the fine-tuned models themselves show distinct generalization behaviors when tested outside the adaptation task's distribution. More specifically, we first show that the weight matrices trained with LoRA have new, high-ranking singular vectors, which we call intruder dimensions. Intruder dimensions do not appear during full fine-tuning. Second, we show that LoRA models with intruder dimensions, despite achieving similar performance to full fine-tuning on the target task, become worse models of the pre-training distribution and adapt less robustly to multiple tasks sequentially. Higher-rank, rank-stabilized LoRA models closely mirror full fine-tuning, even when performing on par with lower-rank LoRA models on the same tasks. These results suggest that models updated with LoRA and full fine-tuning access different parts of parameter space, even when they perform equally on the fine-tuned distribution. We conclude by examining why intruder dimensions appear in LoRA fine-tuned models, why they are undesirable, and how their effects can be minimized.

Parameter-Efficient Fine-Tuning (PEFT) has become an essential approach for adapting large-scale pre-trained models while reducing computational costs. Among PEFT methods, LoRA significantly reduces trainable parameters by decomposing weight updates into low-rank matrices. However, traditional LoRA applies a fixed rank across all layers, failing to account for the varying complexity of hierarchical information, which leads to inefficient adaptation and redundancy. To address this, we propose MSPLoRA (Multi-Scale Pyramid LoRA), which introduces Global Shared LoRA, Mid-Level Shared LoRA, and Layer-Specific LoRA to capture global patterns, mid-level features, and fine-grained information, respectively. This hierarchical structure reduces inter-layer redundancy while maintaining strong adaptation capability. Experiments on various NLP tasks demonstrate that MSPLoRA achieves more efficient adaptation and better performance while significantly reducing the number of trainable parameters. Furthermore, additional analyses based on Singular Value Decomposition validate its information decoupling ability, highlighting MSPLoRA as a scalable and effective optimization strategy for parameter-efficient fine-tuning in large language models. Our code is available at https://github.com/Oblivioniss/MSPLoRA.

Low-rank optimization has emerged as a promising direction in training large language models (LLMs) to reduce the memory usage of adaptive optimizers by constraining learning to a lower-dimensional space. Prior work typically projects gradients of linear layers using approaches based on Singular Value Decomposition (SVD). However, applying SVD-based procedures individually to each layer in large models is computationally expensive and incurs additional memory costs due to storing the projection matrices. In this work, we propose a computationally efficient and conceptually simple two-step procedure to approximate SVD-based gradient projections into lower-dimensional spaces. First, we construct a complete orthogonal basis using predefined orthogonal matrices of the Discrete Cosine Transform (DCT). Second, we adaptively select basis columns based on their alignment with the gradient of each layer. Each projection matrix in our method is obtained via a single matrix multiplication followed by a lightweight sorting step to identify the most relevant basis vectors. Due to the predefined nature of the orthogonal bases, they are computed once at the start of training. During training, we store only the indices of the selected columns, avoiding the need to store full projection matrices for each layer. Our numerical experiments on both pre-training and fine-tuning tasks demonstrate the effectiveness of our dual strategy in approximating optimal low-rank projections, matching the performance of costly SVD-based methods while achieving faster runtime and reduced memory usage.

Model merging enables the combination of multiple specialized expert models into a single model capable of performing multiple tasks. However, the benefits of merging an increasing amount of specialized experts generally lead to diminishing returns and reduced overall performance gains. In this work, we empirically and theoretically analyze this limitation, proving that for Task Arithmetic-based methods, as more experts are merged, the common information dominates the task-specific information, leading to inevitable rank collapse. To mitigate this issue, we introduce Subspace Boosting, which operates on the singular value decomposed task vector space and maintains task vector ranks. Subspace Boosting raises merging efficacy for up to 20 experts by large margins of more than 10% when evaluated on both vision and language benchmarks. Moreover, we propose employing Higher-Order Generalized Singular Value Decomposition to quantify task similarity, offering a new interpretable perspective on model merging. Code and models are available at https://github.com/ronskoro/Subspace-Boosting.

Model compression is key to mitigate deployment challenges of ever growing machine learning models. In this area of research, singular value decomposition (SVD)-based compression offers a compelling trade-off between computational efficiency and model accuracy. Fisher-weighted SVD in particular provides principled, loss-aware compression. However, we find that improving the fidelity of Fisher approximation used in the compression is poorly predictive of post-compression accuracy for Vision Transformers (ViTs). Motivated by this observation, we propose FACTS, a structured Fisher Approximation tailored to Compressing ViTs with Fisher-weighted SVD, which enforces token-local aggregation while preserving within-token activation-gradient dependence. Additionally, we introduce a fast Constrained Rank Search (CoRS), that optimizes layer-wise rank allocation while adhering to a fixed floating point operation (FLOP) constraint. Extensive experiments across ViTs and hybrid architectures demonstrate that FACTS consistently improves accuracy-efficiency trade-offs without requiring finetuning. Notably, it outperforms the strongest SVD baseline by up to +5.8 percentage points (p.p.) Top-1 on Swin-B, with further gains driven by our search method. Code is available at https://github.com/MoritzTho/FACTS.

Mitigating catastrophic risks from large language models requires excising hazardous operational capabilities, such as cyberwarfare exploitation and pathogen enhancement, without degrading benign scientific utility. In these dual-use domains, legitimate defensive science and malicious attacks share vocabulary, syntax, and internal model representations, causing standard unlearning methods to induce severe collateral damage (concept bleeding). In this work, we present Orthogonal Boundary Unlearning (OBU), a mathematically grounded framework designed to eliminate collateral damage on benign dual-use capabilities. OBU extracts a domain-matched capability subspace from paired calibration queries via singular value decomposition on empirical activation covariances, and restricts parameter updates strictly to the orthogonal null space. This guarantees that benign representations experience zero parameter perturbation along the spanned calibration subspace. We present two complementary realizations of OBU: Projected LoRA, which applies the orthogonal constraint during iterative gradient-based fine-tuning, and Closed-Form Weight Surgery, which directly edits model weights in a single analytical step without training. Across four modern open-weight foundation models (Gemma-3-4B, Qwen3.5-4B, SmolLM3-3B, and Granite-4.2-3B), OBU achieves strong dual-use concept separation, substantially mitigates in-context priming attacks, and demonstrates durable resistance against adversarial relearning without degrading benign domain competence.

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

Doubly-stochastic matrices (DSM) are increasingly utilized in structure-preserving deep architectures -- such as Optimal Transport layers and Sinkhorn-based attention -- to enforce numerical stability and probabilistic interpretability. In this work, we identify a critical spectral degradation phenomenon inherent to these constraints, termed the Homogeneity Trap. We demonstrate that the maximum-entropy bias, typical of Sinkhorn-based projections, drives the mixing operator towards the uniform barycenter, thereby suppressing the subdominant singular value σ_2 and filtering out high-frequency feature components. We derive a spectral bound linking σ_2 to the network's effective depth, showing that high-entropy constraints restrict feature transformation to a shallow effective receptive field. Furthermore, we formally demonstrate that Layer Normalization fails to mitigate this collapse in noise-dominated regimes; specifically, when spectral filtering degrades the Signal-to-Noise Ratio (SNR) below a critical threshold, geometric structure is irreversibly lost to noise-induced orthogonal collapse. Our findings highlight a fundamental trade-off between entropic stability and spectral expressivity in DSM-constrained networks.

Model merging aims to integrate multiple task-adapted models into a unified model that preserves the knowledge of each task. In this paper, we identify that the key to this knowledge retention lies in maintaining the directional consistency of singular spaces between merged multi-task vector and individual task vectors. However, this consistency is frequently compromised by two issues: i) an imbalanced energy distribution within task vectors, where a small fraction of singular values dominate the total energy, leading to the neglect of semantically important but weaker components upon merging, and ii) the geometric inconsistency of task vectors in parameter space, which causes direct merging to distort their underlying directional geometry. To address these challenges, we propose DC-Merge, a method for directional-consistent model merging. It first balances the energy distribution of each task vector by smoothing its singular values, ensuring all knowledge components are adequately represented. These energy-balanced vectors are then projected onto a shared orthogonal subspace to align their directional geometries with minimal reconstruction error. Finally, the aligned vectors are aggregated in the shared orthogonal subspace and projected back to the original parameter space. Extensive experiments on vision and vision-language benchmarks show that DC-Merge consistently achieves state-of-the-art performance in both full fine-tuning and LoRA settings. The implementation code is available at https://github.com/Tobeginwith/DC-Merge.

We consider the convergence of the ESD for non-Hermitian random band matrices with independent entries to the circular law, which is the uniform measure on the unit disk in the center of the complex plane. We assume that the bandwidth of the matrix scales like n^γ for some γin(0,1], where n is the matrix size, and the variance profile of the matrix is only assumed to be doubly stochastic with no additional assumption on its specific mixing properties. We prove that the circular law limit holds either (1) when γ>5{6} and the entries are independent Gaussians, (2) or when γ>8{9} and the entries are independent subgaussian random variables. This new threshold improves the previous threshold γ>32{33} which was only proven for block band matrices and periodic band matrices. After the initial version of this paper, the author further extended the range of circular law for much smaller values of γ in 2508.18143 and 2511.01744 when the variance profile has specific mixing properties, but not for an arbitrary doubly stochastic variance profile. Thus the main contribution of this paper is the circular law for a genuine power law bandwidth for any doubly stochastic variance profile. We also prove an extended form of product circular law with a growing number of matrices. Weak delocalization estimates on eigenvectors are also derived. The new technical input is new polynomial lower bounds on some intermediate small singular values, and this estimate does not depend on the specific structure of the variance profile beyond the fact that it is doubly stochastic.

Current methods for large language model alignment typically use scalar human preference labels. However, this convention tends to oversimplify the multi-dimensional and heterogeneous nature of human preferences, leading to reduced expressivity and even misalignment. This paper presents Panacea, an innovative approach that reframes alignment as a multi-dimensional preference optimization problem. Panacea trains a single model capable of adapting online and Pareto-optimally to diverse sets of preferences without the need for further tuning. A major challenge here is using a low-dimensional preference vector to guide the model's behavior, despite it being governed by an overwhelmingly large number of parameters. To address this, Panacea is designed to use singular value decomposition (SVD)-based low-rank adaptation, which allows the preference vector to be simply injected online as singular values. Theoretically, we prove that Panacea recovers the entire Pareto front with common loss aggregation methods under mild conditions. Moreover, our experiments demonstrate, for the first time, the feasibility of aligning a single LLM to represent a spectrum of human preferences through various optimization methods. Our work marks a step forward in effectively and efficiently aligning models to diverse and intricate human preferences in a controllable and Pareto-optimal manner.

Minor Component Adaptation (MiCA) is a novel parameter-efficient fine-tuning method for large language models that focuses on adapting underutilized subspaces of model representations. Unlike conventional methods such as Low-Rank Adaptation (LoRA), which target dominant subspaces, MiCA leverages Singular Value Decomposition to identify subspaces related to minor singular vectors associated with the least significant singular values and constrains the update of parameters during fine-tuning to those directions. This strategy leads to up to 5.9x improvement in knowledge acquisition under optimized training hyperparameters and a minimal parameter footprint of 6-60% compared to LoRA. These results suggest that constraining adaptation to minor singular directions provides a more efficient and stable mechanism for integrating new knowledge into pre-trained language models.

Emergent misalignment (EM) is a recently discovered phenomenon in LLMs where fine-tuning on a narrow misaligned task, such as writing insecure code, leads to broadly misaligned behaviour on unrelated prompts. Previous work has noted that the severity of EM is highly sensitive to training choices; however, we still lack a systematic characterisation of this sensitivity. We perform a sweep over several Qwen3 models, optimisers, datasets, and batch sizes, and find that the choice of optimiser has the largest effect, producing a 7x spread in misalignment rate. Surprisingly, model size has a negligible effect within the Qwen3 family. An additional sweep over 12 models from three families using Adam confirms that model scale (1B-235B) and family have negligible effects for that optimiser. Analysing the loss-alignment relationship on Qwen3-8B, we find that final log training loss is a strong predictor of alignment, and that stratifying by optimiser captures nearly all the residual variance. Training dynamics reveal that each optimiser follows a different trajectory through loss-alignment space, and that after significant training, the optimiser becomes more important than training loss as a predictor of alignment. Muon, the adaptive optimiser that preserves alignment the best, implicitly regularises for a more uniform distribution of singular values of the LoRA adapter. We evaluate this insight by training with an additional loss term that incentivises a flatter singular value spectrum, and find that this substantially recovers alignment for the more EM-prone adaptive optimisers (Adam and Lion), with negligible cost to training loss. These results identify optimiser choice as a key factor in EM severity, but show that spectral regularisation can substantially mitigate the effects of EM-prone optimisers.

Short-recurrence Krylov methods that are equivalent in exact arithmetic often diverge in floating-point arithmetic. To illustrate this, we provide an exactly representable two-cycle for steepest descent with a recursively updated residual. A complementary convergence theorem gives a sufficient condition under which the stored residual decreases geometrically. We then show that a second positive definite family yields different outcomes for a Hestenes--Stiefel Conjugate Gradient (CG) implementation and a direct Lanczos--Galerkin approach. CG finds the exact solution after four updates, whereas the two-step projected system becomes inconsistent. Under stated perturbation and transfer hypotheses, a common spectral enclosure gives comparable convergence bounds. For a dense, left-to-right evaluation order, we prove a sufficient precision bound for a prescribed backward error within n updates for an ntimes n matrix, together with a computable stopping test and specified exponent-range assumptions. We use polynomial estimates and numerical experiments to examine the trade-off between working precision and the number of CG iterations needed to achieve a prescribed backward error. We also compare results on inexact matrix-vector products and preconditioning with the frameworks of Paige and Greenbaum. For block Lanczos, we analyze Householder orthogonalization and singular-value truncation as the block size changes. Under componentwise and normwise error bounds, the computed coefficients satisfy a controlled local recurrence and an exact block Lanczos relation for a nearby symmetric problem in a larger space. A block form of Paige's identity bounds the overlap with a Ritz vector along its residual coordinate direction. An inter-block recurrence describes the evolution of overlap, and a Gram-matrix argument gives a count of additional nearby Ritz values.

Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy enables parameter sharing and efficient reconstruction and has been widely adopted across domains ranging from multi-view learning and signal processing to neural network compression. However, it leaves a fundamental question unanswered: which matrices can be safely concatenated and compressed together under explicit reconstruction error constraints? Existing approaches rely on heuristic or architecture-specific grouping and provide no principled guarantees on the resulting SVD approximation error. In the present work, we introduce a theory-driven framework for compression-aware clustering of matrices under SVD compression constraints. Our analysis establishes new spectral bounds for horizontally concatenated matrices, deriving global upper bounds on the optimal rank-r SVD reconstruction error from lower bounds on singular value growth. The first bound follows from Weyl-type monotonicity under blockwise extensions, while the second leverages singular values of incremental residuals to yield tighter, per-block guarantees. We further develop an efficient approximate estimator based on incremental truncated SVD that tracks dominant singular values without forming the full concatenated matrix. Therefore, we propose three clustering algorithms that merge matrices only when their predicted joint SVD compression error remains below a user-specified threshold. The algorithms span a trade-off between speed, provable accuracy, and scalability, enabling compression-aware clustering with explicit error control. Code is available online.

Beam codebooks are a new feature of massive multiple-input multiple-output (M-MIMO) in 5G new radio (NR). Codebooks comprised of beamforming vectors are used to transmit reference signals and obtain limited channel state information (CSI) from receivers via the codeword index. This enables large arrays that cannot otherwise obtain sufficient CSI. The performance, however, is limited by the codebook design. In this paper, we show that machine learning can be used to train site-specific codebooks for initial access. We design a neural network based on an autoencoder architecture that uses a beamspace observation in combination with RF environment characteristics to improve the synchronization signal (SS) burst codebook. We test our algorithm using a flexible dataset of channels generated from QuaDRiGa. The results show that our model outperforms the industry standard (DFT beams) and approaches the optimal performance (perfect CSI and singular value decomposition (SVD)-based beamforming), using only a few bits of feedback.

Large language models (LLMs) show excellent performance in difficult tasks, but they often require massive memories and computational resources. How to reduce the parameter scale of LLMs has become research hotspots. In this study, we make an important observation that the multi-head self-attention (MHA) sub-layer of Transformer exhibits noticeable low-rank structure, while the feed-forward network (FFN) sub-layer does not. With this regard, we design a mixed compression model, which organically combines Low-Rank matrix approximation And structured Pruning (LoRAP). For the MHA sub-layer, we propose an input activation weighted singular value decomposition method to strengthen the low-rank characteristic. Furthermore, we discover that the weight matrices in MHA sub-layer have different low-rank degrees. Thus, a novel parameter allocation scheme according to the discrepancy of low-rank degrees is devised. For the FFN sub-layer, we propose a gradient-free structured channel pruning method. During the pruning, we get an interesting finding that the least important 1% of parameter actually play a vital role in model performance. Extensive evaluations on zero-shot perplexity and zero-shot task classification indicate that our proposal is superior to previous structured compression rivals under multiple compression ratios.

The recent trend in scaling language models has led to a growing demand for parameter-efficient tuning (PEFT) methods such as LoRA (Low-Rank Adaptation). LoRA consistently matches or surpasses the full fine-tuning baseline with fewer parameters. However, handling numerous task-specific or user-specific LoRA modules on top of a base model still presents significant storage challenges. To address this, we introduce LoRA-XS (Low-Rank Adaptation with eXtremely Small number of parameters), a novel approach leveraging Singular Value Decomposition (SVD) for parameter-efficient fine-tuning. LoRA-XS introduces a small r x r weight matrix between frozen LoRA matrices, which are constructed by SVD of the original weight matrix. Training only r x r weight matrices ensures independence from model dimensions, enabling more parameter-efficient fine-tuning, especially for larger models. LoRA-XS achieves a remarkable reduction of trainable parameters by over 100x in 7B models compared to LoRA. Our benchmarking across various scales, including GLUE, GSM8k, and MATH benchmarks, shows that our approach outperforms LoRA and recent state-of-the-art approaches like VeRA in terms of parameter efficiency while maintaining competitive performance.

Recently, a wide range of memory-efficient LLM training algorithms have gained substantial popularity. These methods leverage the low-rank structure of gradients to project optimizer states into a subspace using projection matrix found by singular value decomposition (SVD). However, convergence of these algorithms is highly dependent on the update rules of their projection matrix. In this work, we provide the first convergence guarantee for arbitrary update rules of projection matrix. This guarantee is generally applicable to optimizers that can be analyzed with Hamiltonian Descent, including most common ones, such as LION, Adam. Inspired by our theoretical understanding, we propose Online Subspace Descent, a new family of subspace descent optimizer without SVD. Instead of updating the projection matrix with eigenvectors, Online Subspace Descent updates the projection matrix with online PCA. Online Subspace Descent is flexible and introduces only minimum overhead to training. We show that for the task of pretraining LLaMA models ranging from 60M to 7B parameters on the C4 dataset, Online Subspace Descent achieves lower perplexity and better downstream tasks performance than state-of-the-art low-rank training methods across different settings and narrows the gap with full-rank baselines.