Singular Value Decomposition

Recently, Muon and related spectral optimizers have demonstrated strong empirical performance as scalable stochastic methods, often outperforming Adam. Yet their behaviour remains poorly understood. We analyze stochastic spectral optimizers, including Muon, on a high-dimensional matrix-valued least squares problem. We derive explicit deterministic dynamics that provide a tractable framework for studying learning behaviour with a focus on (stochastic) SignSVD, which Muon approximates, and (stochastic) SignSGD, the latter serving as a proxy for Adam. Our analysis shows that for large batch size, SignSVD performs a square-root preconditioning with respect to the data covariance spectrum, while for small batch size smaller eigenmodes behave like SGD, slowing down convergence. We contrast with SignSGD which for generic covariance performs no preconditioning and has no transition, leading to different optimal learning rates and convergence characteristics. The two methods match up to a constant factor with isotropic data, but behave differently with anisotropic data. An analysis of a power law covariance model with data exponent $α$ and target exponent $β$ shows there are three phases in the $(α,β)$ plane: one where SignSGD is uniformly favored, one where SignSVD is uniformly favored, and a third where the two methods exhibit a trade-off in performance.

This paper relaxes the restrictive symmetry conditions adopted in [4], [5] and extends their universal feature selection framework to accommodate noisy observations as well as attribute structures that may exhibit directional preferences. We introduce the notion of weak spherical symmetry, quantified by second-moment distances, which allows controlled deviations from rotational invariance. Under this relaxed condition, we develop a universal feature selection framework based on the singular value decomposition of the canonical dependence matrix computed from noisy data. Our main result shows that the selected features achieve asymptotically optimal error exponents up to a residual term that depends on the symmetry deviation $δ$ and the noise levels $η_1, η_2$. When $δ, η_1, η_2$ are relatively small, our result recovers that of [5], thereby demonstrating that exact spherical symmetry is unnecessary. Overall, our findings highlight the robustness of the selection framework against second-moment deviations and observation noise, thereby broadening its applicability across diverse inference tasks and providing a theoretically grounded tool for universal feature selection in practical scenarios.

SVD-based Low-rank compression reduces transformer parameters and nominal FLOPs, but these savings often translate poorly into real LLM serving speedups. We show that this gap is largely a runtime problem: factorized checkpoints fragment execution paths, and the resulting overhead differs substantially between prefill and autoregressive decode. We present FlashSVD v1.5, a unified inference runtime for serving SVD-compressed transformers. FlashSVD v1.5 maps diverse public SVD compression families to a common factorized representation and combines phase-specific kernels with dense-KV decode, packed MLP execution, and per-layer CUDA-graph replay to reorganize the low-rank serving path into a thin runtime. Across representative decoder-serving settings, FlashSVD v1.5 achieves up to 2.55x decode and 2.39x end-to-end speedup, and it attains 1.48x average decode and 1.44x average end-to-end speedup across multiple popular SVD compression families. These results suggest that practical low-rank acceleration requires runtime co-design, not compression algorithms alone. Our code is available at: https://github.com/Zishan-Shao/FlashSVD.

Random-feature neural networks (RFNNs), including architectures with fixed hidden layers and analytically determined output weights, offer fast training but often suffer from issues due to dense representations of the hidden layer activation. Their reliance on dense feature mappings and least squares solvers can limit scalability and numerical stability, particularly for high-dimensional or stiff systems. Specifically, the activation matrix is observed to be low-rank and extremely ill-conditioned. In this work, we propose a sparse framework for RFNNs that integrates structured sparsity into the hidden layer activations that increases the rank and employs Sparse Singular Value Decomposition (sSVD) for solving the resulting linear least squares problem scalably and efficiently while catering to the bad condition number. We explore the theory behind Lanczos-Golub-Kahan Bidiagonalization technique for sparse SVD and conduct some experiments to identify some limitations and justify the requirement for orthogonalization step in our application. Then, we demonstrate that the proposed method maintains or improves solution accuracy for solving the benchmark one-dimensional steady convection-diffusion equations case having stronger advection, while achieving substantial gains in training efficiency and robustness compared to standard dense implementations.

How few parameters do we really need to forecast a periodic time series? An hourly electricity series, reshaped as a 24-row matrix with one column per day, is approximately rank-1: a daily shape modulated by a daily level (median centered rank-1 energy 0.82 on GIFT-Eval). Should we learn the shape? Smoothing, shrinkage, and low-rank fits all seem like obvious upgrades over the simple average of the last K=2 cycles. On all 97 GIFT-Eval configurations, we tested 8 such alternatives (e.g., Fourier, EWMA, James-Stein, rank-r SVD): none significantly beats the frozen baseline under Holm correction; two are significantly worse. The resulting method, FLAIR, is (a) Effective: matches PatchTST on aggregate GIFT-Eval (relMASE 0.838 vs 0.849); (b) Compact: 28 scalars for hourly, 57 for weekly; (c) Fast: 22 minutes on one CPU core of a MacBook Pro; (d) Closed-form & Hands-Off: one SVD per period candidate, GCV-averaged Ridge, no GPU, no pre-training, no per-task tuning. In the high-rank-1, many-cycle regime, extra flexibility is estimation noise.

Building on the abstract Generalized Singular Value Decomposition (GSVD) theory of Brown et al. [2025], we prove that most modern neural architectures admit a generalized SVD representation in which they are left-invertible before a final linear layer, with no change in input-output behavior. Furthermore, the left-invertible nonlinear portion of the input-output behavior can be made to be \emph{norm preserving}, meaning that perturbations in the left-invertible ``embedding'' (the activations prior to the final linear layer in this representation) correspond proportionally to changes in the input, i.e., distance in feature space can be calibrated directly to distance in input space. We provide a data-driven algorithm for estimating this representation from trained models and propose a model architecture that naturally facilitates the decomposition. We then provide a proof-of-concept that the learned representation can be used to identify adversarial perturbations to model inputs, and develop the theory necessary for future applications to areas such as model bias and invertibility.

Perceptual image compression focuses on preserving high visual quality under low-bitrate constraints. Most existing approaches to perceptual compression leverage the strong generative capabilities of generative adversarial networks or diffusion models, at the cost of substantial model complexity. To this end, we present an efficient perceptual image compression method that exploits the long-range modeling capability and linear computational complexity of state space models, with a particular focus on Mamba. Unlike existing methods that rely on an inherently fixed scanning order and consequently impair semantic continuity and spatial correlation, we develop a semantic-aware Mamba block (SAMB) to enable scanning guided by dynamically clustered semantic features, thereby alleviating the strict causality constraints and long-range information decay inherent to Mamba. Inspired by singular value decomposition, we design an SVD-inspired redundancy reduction module (SVD-RRM) that performs a low-rank approximation on the latent features by introducing a learnable soft threshold, leading to channel-wise redundancy information reduction. The proposed SAMB is integrated into both the encoder and decoder of the compression framework, whereas the SVD-RRM is incorporated only in the encoder. Extensive experiments demonstrate that our method performs favorably against state-of-the-art approaches in terms of rate-distortion-perception tradeoff and model complexity. The source code and pretrained models will be available at https://github.com/Jasmine-aiq/SAMIC.

Modern deep learning usually treats models as separate artifacts: trained independently, specialized for particular purposes, and replaced when improved versions appear. This thesis studies model merging as an alternative paradigm: combining independently trained neural networks directly in weight space, with little or no optimization and without requiring access to the original training data. The thesis considers two main regimes. In the single-task setting, where models share an objective but differ in initialization, we introduce C$^2$M$^3$, a cycle-consistent merging algorithm based on Frank-Wolfe optimization. C$^2$M$^3$ aligns multiple networks into a shared, reference-free parameter space, making weight averaging meaningful without privileging any individual model. In the multi-task setting, where models are fine-tuned for different downstream tasks from a common pretrained initialization, we first develop a theoretical account of task vectors as approximate gradients. This explains both the effectiveness and the limitations of task arithmetic. Building on this view, we show that task vectors inherit the low-rank structure of gradients and introduce Task Singular Vectors (TSV), a decomposition that enables compression and interference reduction through TSV-Merge. We then present MASS, an input-adaptive routing method that uses TSV geometry to select task-relevant subspaces at inference time. Finally, we introduce MERGE$^3$, an evolutionary merging framework that uses Item Response Theory to reduce evaluation costs by up to 50$\times$ while preserving solution quality. Together, these contributions provide theoretical and algorithmic foundations for model merging, supporting a paradigm in which learned capabilities can be composed, reused, and extended across models.

EigenDecomposition (ED) is at the heart of many computer vision algorithms and applications. One crucial bottleneck limiting its usage is the expensive computation cost, particularly for a mini-batch of matrices in deep neural networks. Our previous work proposed a dedicated QR-based ED algorithm for batched small matrices (dim${<}32$). This short paper targets the limitation and proposes a batch-efficient Divide-and-Conquer based ED algorithm for larger matrices. The numerical test shows that for a mini-batch of matrices whose dimensions are smaller than $64$, our method can be much faster than the Pytorch SVD function.

We show that replacing the rolling SVD of AdamW updates with a rolling SVD of loss gradients changes the diagnostic by 1-2 orders of magnitude. Performing SVD on the loss gradient instead of the AdamW update increases the measured perturbative coupling between SED directions and Linear Centroid Hypothesis (LCH) features from $\bar{R}_k \approx 3$--$9\times$ to $100$--$330\times$ across four single-task modular arithmetic operations, eliminating the apparent operation dependence in the original measurement. On a multitask transformer with a shared encoder, update-based SED gives $\bar{R}_k \leq 1$ -- an apparent failure of the diagnostic -- while per-operation gradient-based SED recovers $\bar{R}_k = 20$--$45\times$ across all four operations. Gradient aggregation across competing tasks is the main obstruction; performing SVD on per-task gradients resolves it. A causal intervention shows that constraining attention updates to any rank-3 subspace (whether SED-derived or random) accelerates grokking by approximately $2.3\times$ across random seeds and operations, while removing the rank-3 component has negligible effect under proper gradient-projection methodology. The SED-LCH coupling is therefore a strong diagnostic of where feature formation concentrates in parameter space, but it is not a unique causal pathway: the natural full-rank AdamW attention update is highly rank-redundant under our hyperparameters.

Federated learning (FL) offers a promising distributed learning paradigm for internet of vehicles (IoV) applications. However, it faces challenges from communication overhead and dynamic environments. Model compression techniques reduce computing and communication burden yet create trade-offs between compression ratios and vehicle participation strategies. In this paper, we propose a lightweight FL algorithm named federated learning with dynamic low-rank adaptation (Fed-DLoRA), which is combined with low-rank adaptation (LoRA) to effectively reduce parameters and communication costs while enhancing training efficiency. The convergence analysis of Fed-DLoRA is conducted through stochastic gradient descent optimization coupled with singular value decomposition. This analysis establishes the theoretical relationships among LoRA rank, vehicular scheduling strategies and the model's convergence characteristics. Building on these insights, we formulate a joint optimization problem aimed at maximizing system performance. To address this problem, we propose an adaptive rank, bandwidth and vehicle selection (ARBVS) algorithm that integrates enumeration with greedy optimization strategies. The algorithm provides efficient rank selection and resource scheduling strategies for each FL communication round, thereby achieving effective performance improvements for the FL system. Experimental results demonstrate that Fed-DLoRA achieves superior performance compared to conventional federated learning approaches, exhibiting enhanced accuracy, faster convergence, and improved communication efficiency.

Convolutional neural networks (CNNs) have become increasingly difficult to deploy in resource-constrained environments due to their large memory and computational requirements. Although low-rank compression methods can reduce this burden, most existing approaches compress spatial and channel redundancy independently and therefore do not fully exploit the localised structure within convolutional feature maps. This paper proposes a hierarchical spatio-channel low-rank compression framework for CNNs that exploits redundancy across spatial regions and channel activations. Unlike conventional methods, which apply a uniform decomposition across an entire layer, the proposed approach first partitions feature maps into spatial regions, then groups channels according to their co-activation patterns within each region, and finally applies rank-adaptive SVD to each resulting spatio-channel cluster. The method is evaluated on an AlexNet-based brain tumour MRI classification model and compared with Global SVD and Tucker decomposition under $3\times$ and $6\times$ compression budgets. Our method outperforms both baselines, reducing FLOPs from $8.21\,\mathrm{G}$ to $1.55\,\mathrm{G}$ ($81.1\%$ reduction), achieving a $1.38\times$ inference speed-up, and increasing classification accuracy from $87.76\%$ to $89.80\%$. The method also improves the macro $F_1$-score and performance on challenging classes such as meningioma. A hyper-parameter trade-off analysis demonstrates that the framework provides Pareto-optimal configurations, enabling control over the balance between compression and predictive performance. Moderate clustering with adaptive rank selection yields strong results. Bootstrap standard errors are reported for all classification metrics.

Estimating the number of components is a fundamental challenge in unsupervised learning, particularly when dealing with high-dimensional data with many components or severely imbalanced component sizes. This paper addresses this challenge for classical Gaussian mixture models. The proposed estimator is simple: center the data, compute the singular values of the centered matrix, and count those above a threshold. No iterative fitting, no likelihood calculation, and no prior knowledge of the number of components are required. We prove that, under a mild separation condition on the component centers, the estimator consistently recovers the true number of components. The result holds in high-dimensional settings where the dimension can be much larger than the sample size. It also holds when the number of components grows to the smaller of the dimension and the sample size, even under severe imbalance among component sizes. Computationally, the method is extremely fast: for example, it processes ten million samples in one hundred dimensions within one minute. Extensive experimental studies confirm its accuracy in challenging settings such as high dimensionality, many components, and severe class imbalance.

Adapting foundation models under resource budgets relies heavily on Parameter-Efficient Fine-Tuning (PEFT), with LoRA being a standard modular solution. However, LoRA suffers from spectral interference. Low-rank updates often concentrate energy on the leading singular directions of pretrained weights, perturbing general capabilities and causing catastrophic forgetting and fragile multi-adapter merging. To resolve this, we propose HiP-LoRA, a spectrum-aware adaptation framework. Utilizing the cached singular value decomposition (SVD) of pretrained layers, HiP-LoRA decomposes updates into two channels: a principal channel within the dominant singular subspace, and a residual low-rank channel in the orthogonal complement. A singular-value-weighted stability budget on the principal channel continuously balances pretrained behavior preservation with task-specific plasticity. Experiments on Llama-3.1-8B demonstrate that under matched budgets, HiP-LoRA drastically reduces pretraining degradation and multi-adapter MergeFail, robustly outperforming baselines in interference-sensitive tasks like continual tuning and knowledge editing.

Recursive architectures such as Tiny Recursive Models (TRMs) perform implicit reasoning through iterative latent computation, yet the geometric structure of these reasoning trajectories remains poorly understood. We investigate the activation manifold of TRMs during recursive unrolling and find that activations occupy an effectively linear, low-dimensional subspace whose principal directions can be tracked dynamically with cheap power iterations. This suggests that weight-sharing concentrates iterative computation along a small number of dominant eigendirections, and we find that this concentration varies sharply across computational sites. We exploit this structure through LASER (Low-Rank Activation SVD for Efficient Recursion), a dynamic compression framework that maintains an evolving low-rank basis via matrix-free subspace tracking with a fidelity-triggered reset mechanism, achieving ${\sim}60\%$ activation memory savings with no statistically significant accuracy degradation. Our analysis raises questions about how recursive architectures allocate representational capacity during implicit reasoning, and whether this concentration can be exploited to improve the efficiency and stability of latent computation.

This work introduces Structured 3D-SVD as a practical framework for the reconstruction, compression, and analysis of biological volumetric data. Inspired by the logic of matrix singular value decomposition (SVD), the proposed approach represents third-order volumetric data in the spatial domain and supports progressive reconstruction through ordered quasi-singular coeffients. The experimental evaluation was carried out on two biological volumetric datasets: one full-volume scan of a fish and another of a brain. The results show that Structured 3D-SVD achieves reconstruction quality close to that of Tucker decomposition while requiring shorter computation times and outperforms canonical polyadic decomposition (CPD) in both accuracy and runtime. In addition, a progressive reconstruction analysis shows that relatively low truncation levels are sufficient to preserve the main volumetric structures, while higher truncation levels lead to more detailed reconstructions.

Vision-Language Models (VLMs) have revolutionized multi-modal learning by jointly processing visual and textual information. Yet, they face significant challenges due to the high computational and memory demands of processing long sequences of vision tokens. Many existing methods rely on local heuristics, such as attention scores or token norms. However, these criteria suffer from positional bias and information dispersion, limiting their ability to preserve essential content at high pruning ratios and leading to performance degradation on visually detailed images. To address these issues, we propose SVD-Prune, a training-free, plug-and-play token pruning method based on Singular Value Decomposition. It decomposes the vision token feature matrix and selects the top-k tokens using statistical leverage scores, ensuring only tokens contributing most to the dominant global variance are preserved. Experiments show that SVD-Prune consistently outperforms prior pruning methods under extreme vision token budgets, maintaining strong performance even with 32 and 16 vision tokens.

Parameter-efficient continual learning aims to adapt pre-trained models to sequential tasks without forgetting previously acquired knowledge. Most existing approaches treat continual learning as avoiding interference with past updates, rather than considering what properties make the current task-specific update naturally preserve previously acquired knowledge. From a knowledge-decomposition perspective, we observe that low-rank adaptations exhibit highly imbalanced singular value spectra: a few dominant components absorb most of the adaptation energy, thereby (i) more likely to disrupt previously acquired knowledge and (ii) making the update more vulnerable to interference from subsequent tasks. To enable explicit balance among components, we decouple the magnitude of the task update from its directional structure and formulate it as a constrained optimization problem on a restricted Stiefel manifold. We address this problem using a projected first-order method compatible with standard deep-learning optimizers used in vision-language models. Our method mitigates both backward and forward forgetting, consistently outperforming continual learning baselines. The implementation code is available at https://github.com/haodotgu/EBLoRA.

Model merging has emerged as a promising paradigm for enabling multi-task capabilities without additional training. However, traditional basic merging methods often experience performance degradation due to parameter conflicts, even when applied to similar tasks. While recent personalized merging frameworks successfully preserve task-specific information to maintain performance, they typically incur storage overhead. In this paper, we propose Decomposition, Thresholding, and Scaling (DTS), an approximation-based personalized merging framework that pushes task-specific storage efficiency. DTS first applies singular value decomposition to the task-specific information and retains only a small subset of singular values and vectors. It then introduces a novel thresholding strategy that partitions singular vector elements into groups and assigns a scaling factor to each group. To enable generalization to unseen tasks, we further extend DTS with a variant that fuses task-specific information in a data-free manner based on the semantic similarity of task characteristics. Extensive experiments demonstrate that DTS consistently outperforms state-of-the-art baselines while requiring only 1% extra storage per task. Furthermore, experiments on unseen tasks show that the DTS variant achieves significantly better generalization performance. Our code is available at https://github.com/krumpguo/DTS.

As deep learning (DL) techniques become integral to various applications, ensuring model fairness while maintaining high performance has become increasingly critical, particularly in sensitive fields such as medical diagnosis. Although a variety of bias-mitigation methods have been proposed, many rely on computationally expensive debiasing strategies or suffer substantial drops in model accuracy, which limits their practicality in real-world, resource-constrained settings. To address this issue, we propose a fairness-oriented low rank factorization (LRF) framework that leverages singular value decomposition (SVD) to improve DL model fairness. Unlike traditional SVD, which is mainly used for model compression by decomposing and reducing weight matrices, our work shows that SVD can also serve as an effective tool for fairness enhancement. Specifically, we observed that elements in the unitary matrices obtained from SVD contribute unequally to model bias across groups defined by sensitive attributes. Motivated by this observation, we propose a method, named FairLRF, that selectively removes bias-inducing elements from unitary matrices to reduce group disparities, thus enhancing model fairness. Extensive experiments show that our method outperforms conventional LRF methods as well as state-of-the-art fairness-enhancing techniques in terms of improving fairness while maintaining accuracy. Additionally, an ablation study examines how major hyper-parameters may influence the performance of processed models. To the best of our knowledge, this is the first work utilizing SVD not primarily for compression but for fairness enhancement of DL classifications.

The Fisher information is a fundamental concept for characterizing the sensitivity of parameters in neural networks. However, leveraging the full observed Fisher information is too expensive for large models, so most methods rely on simple diagonal approximations. While efficient, this approach ignores parameter correlations, often resulting in reduced performance on downstream tasks. In this work, we mitigate these limitations and propose Generalized Fisher-Weighted SVD (GFWSVD), a post-training LLM compression technique that accounts for both diagonal and off-diagonal elements of the Fisher information matrix, providing a more accurate reflection of parameter importance. To make the method tractable, we introduce a scalable adaptation of the Kronecker-factored approximation algorithm for the observed Fisher information. We demonstrate the effectiveness of our method on LLM compression, showing improvements over existing compression baselines. For example, at a 20 compression rate on the MMLU benchmark, our method outperforms FWSVD, which is based on a diagonal approximation of the Fisher information, by 5 percent, SVD-LLM by 3 percent, and ASVD by 6 percent compression rate.

We consider the effect of Gaussian perturbations on least-squares residuals, orthogonal projections, and QR-type algorithms. The problem that motivated our investigations is as follows: suppose that a full column-rank matrix $B\in\mathbb{R}^{m\times n}$ has already been computed, and suppose that a new normalized column $q=(x+y)/\|x+y\|_2$ is to be appended to $B$, where $x\perp\operatorname{span}(B)$ is the ideal orthogonal component and $y$ represents the orthogonalization error. How large can the condition number $κ([B,q])$ of the resulting matrix $[B,q]$ become? While we provide a Weyl-type bound on the singular values of $[B,q]$, in terms of the extremal singular values of $B$ and the quantity $\|B^T y\|_2/\|x+y\|_2$, we also derive exact probability laws for norms and projection residuals under Gaussian perturbations. Finally, we use these probability laws to derive probabilistic condition-number bounds for QR-type processes with imperfect orthogonalization and exact normalization.