Singular Value Decomposition

Also known as SVD

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6 papers in the last four weeks, against 1 the four weeks before. 0.1% of all new papers.

Jul 13Week of Sep 28

Latest papers 26

Oct 7, 2026cs.LG

D-SLR: The Disjoint Row-Sparse plus Low-Rank Decomposition

Compressing a matrix for reconstruction still defaults to the truncated SVD, approximating the data with a single low-rank structure. It is common to reduce the residual further by adding an overlapping row-sparse component, but methods that solve this joint problem often require iterative solvers and tuning of regularization parameters. We propose the Disjoint Row-Sparse plus Low-Rank (D-SLR) decomposition, a closed-form drop-in for the truncated SVD that improves or exactly matches it. D-SLR restricts rows to either being stored verbatim or approximated by the low-rank fit, never both. Under squared error this restriction costs nothing: the joint optimum is attainable disjointly with fewer parameters at every non-trivial rank and stored row count (shape). With zero stored rows D-SLR reduces to the truncated SVD, so it never does worse at equal cost. The algorithm scores the entire error-versus-parameters tradeoff, and the solution is chosen afterwards by a supplied error target or parameter count, or by a selection rule. The grid and solution together cost three SVDs, with no tuning or regularization. We derive an assumption-free, a-posteriori lower bound on the error at every shape, giving each solution a computable certificate on the potential gain of any other choice of rank and stored rows. Experiments on synthetic and real data (LLM embedding tables, network traffic, hyperspectral images) confirm the gains and quantify the certificate.

Vincent Szolnoky

Department of Mathematical Sciences Chalmers University of Technology

Low-Rank Matrix Decomposition Low-Rank Compression Low-Rank Approximation Singular Value Decomposition

Oct 6, 2026cs.LG

Singular Value Decomposition: A Geometric Rediscovery, Where Proofs Become Algorithms

This article is a geometric rediscovery of the singular value decomposition, with a further claim: the construction it builds is the machinery behind much of machine learning. The same argument that answers an idle question about ellipses is the algorithm behind principal component analysis, kernel methods, and PageRank, and it is not only the results that transfer but the proofs themselves, run as procedures. The usual introduction states

A = UΣV^T

and justifies it via the spectral theorem applied to

A^T A

. This is correct but unilluminating, since it assumes a powerful theorem to reach a result that is, in the end, about ellipses. Part I reverses the order. A linear map sends the unit circle to an ellipse; one asks which input directions map to its axes, and finds, example after example, that they are perpendicular. In the plane this can be watched: rotate a frame, track how far its images are from perpendicular, and a sign change forces a frame where they are exactly perpendicular, which is also where the map stretches hardest. Maximizing the stretch and recursing generalizes this to n dimensions, with singular values falling out in order, and the construction proves the spectral theorem rather than assuming it. Part II puts each construction to work: maximize-and-recurse becomes the power method and PageRank; the lemma locating the maximizer becomes the stopping rule of gradient descent; the duality between

A^T A

and

A A^T

becomes the transport at the heart of kernel PCA. Each connection is stated with its boundary, saying what the decomposition supplies and where another idea takes over. Prerequisites are the standard sophomore sequence, and the worked examples are small enough to check by hand.

Paul Agron

Singular Value Decomposition

Sep 21, 2026cs.LG

Prescriptive SVD-Inspired Attention via Spectral Energy Retention

Self-attention is central to modern Transformer architectures, but its dense dot-product formulation makes it difficult to identify which internal directions are structurally important and which can be modified without disrupting the model. SVD-Inspired Attention (SVDA) addresses part of this problem by introducing a learned diagonal spectrum into the query-key score interaction, making latent attention directions explicitly inspectable through indicators such as spectral entropy, effective rank, sparsity, alignment, selectivity, and perturbation response. This paper examines the transition from diagnostic interpretation to operational intervention. A diagnosis--intervention--verification framework is proposed, and one intervention is evaluated: spectral energy retention in the attention-score pathway. Across FashionMNIST, CIFAR-10, CIFAR-100, and Food-101, the

ρ=0.90

prescription removes 24.5--53.7% of score directions, reduces parameters by 2.6--4.3%, and reduces estimated MACs by 2.8--5.4%. The paired mean accuracy change of the dimension-reduced model ranges from

-0.03

to

+0.05

percentage points over three seeds. These results support SVDA as an intrinsically interpretable attention mechanism whose learned spectrum exposes an operational coordinate system for deterministic and verifiable modification of attention-score formation.

Vasileios Arampatzakis, Vasileios Sevetlidis, George Pavlidis

Athena Research Center, Greece

Transformer Interpretability Low-Rank Attention Self-Attention Singular Value Decomposition Neural Network Pruning

Sep 16, 2026cs.LG

Randomized SVD Approximations for Spectral Co-Clustering of Word-Document Matrices

Spectral co-clustering is a useful tool for discovering latent structure in word-document matrices, but its reliance on singular value decomposition (SVD) can make standard formulations expensive on high-dimensional data. This paper presents two randomized approximations for normalized spectral co-clustering of bipartite text data when the numbers of document and word clusters may differ. The first method uses randomized SVD through random projection, while the second combines partial SVD with element-wise random sampling. Across real-world and synthetic datasets, both methods reduce runtime relative to the full-SVD baseline, but their behavior depends on matrix sparsity. The random projection method is the more reliable approximation across the tested settings, whereas the sampling-based method is most useful on denser matrices and provides limited benefit on already sparse text data. These results show that randomized approximations for spectral co-clustering should be selected according to the underlying structure of the data.

Fateme Mazdarani, Carlos Toxtli

School of Computing Clemson University Clemson, SC, USA

Clustering Randomized Sketching Spectral Clustering Singular Value Decomposition

Sep 14, 2026cs.LG

Per-Matrix Optimality Is Not Enough: Three-Level Optimization for Low-Rank LLM Compression

Per-matrix singular value decomposition (SVD) truncation is Eckart-Young optimal in the whitened Frobenius norm, but errors from independently compressed matrices compound through the block's nonlinear forward pass. Inspired in part by hierarchical variational optimization in quantum many-body methods, we introduce a three-level chain that widens optimization scope from individual matrices to Transformer blocks to the full model: whitened SVD~(L1), block-level joint optimization~(L2), and end-to-end language-modeling loss refinement~(L3), all from 256 calibration sequences, with no instruction or recovery data. On LLaMA-7B at 60% compression, the chain reduces WikiText-2 perplexity from 42.1 to 19.1 to 11.4. The block-level stage acts as a regularizer: skipping it worsens Penn Treebank (PTB) perplexity by 24 points, a gap that additional end-to-end training did not close in our experiments. Perplexity gains hold across 20-80% compression, five architectures up to 13B parameters, and both in-distribution and out-of-distribution benchmarks, though the cross-architecture rows use architecture-specific configurations and the ratio sweep was not run under one common protocol. With more calibration data, skipping the block-level stage becomes competitive, revealing an offline compute--data trade-off. We therefore claim improvements only in perplexity and compression fidelity; downstream accuracy remains well below the dense model.

Huicheng Zhang, Xiyao Feng, Ze-Tong Li +6

Hong Kong University of Science and Technology (Guangzhou), Guangdong 511453, China · QudeLeap Research, Shanghai 200030, China

Model Compression LLM Compression Low-Rank Matrix Decomposition Low-Rank Compression Singular Value Decomposition

Sep 13, 2026stat.ML

A Functional SVD Framework for Regularized Multivariate Functional PCA with Dual Penalization

This paper introduces a novel framework for Regularized Multivariate Functional Principal Component Analysis (ReMFPCA) via Functional Singular Value Decomposition (SVD). The proposed method extends existing MFPCA approaches by incorporating a generalized functional SVD within a Hilbert space framework, enabling simultaneous regularization of both functional principal components (PCs) and their associated PC scores. A key innovation of this framework is the inclusion of a sparsity penalty on the PC scores, which enhances interpretability by filtering out irrelevant subject-specific variations. This dual-penalization strategy represents a significant advancement beyond existing covariance-based eigen decomposition methods, which penalize only the functional PCs. Two power algorithm implementations, sequential and joint, are proposed, together with a cross-validation approach based on iterative regression for optimal smoothing parameter selection. Comprehensive simulation studies and real data applications demonstrate that the proposed framework substantially improves the extraction of informative and interpretable components, offering methodological and practical benefits for analyzing multivariate functional data across diverse domains.

Yue Zhao, Hossein Haghbin, Rebecca Sanders +1

Division of Biostatistics and Health Data Science, University of Minnesota, USA · Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Iran · Department of Mathematical and Statistical Sciences, Marquette University, USA

Principal Component Analysis Singular Value Decomposition

Sep 11, 2026cs.LG

Semi-Tensor Product-Based Multi-Term Randomized T-SVD and Its Visual Applications

Tensor singular value decomposition (T-SVD), which is built upon the tensor-tensor product (t-product), has emerged as a powerful tool for processing high-dimensional visual data such as color images and videos. However, the standard t-product imposes strict dimensional compatibility constraints. Although extensions based on the semi-tensor product (STP) relax this restriction, their single-term formulations still suffer from limited approximation accuracy. Moreover, these deterministic methods incur high computational costs when processing large-scale tensor data. To address these issues, this paper introduces a novel semi-tensor product for third-order tensors under the t-product framework induced by arbitrary invertible linear transforms. The resulting tensor semi-tensor product breaks the rigid dimension matching requirement of the standard t-product, while retaining the closed-form property of T-SVD. Based on this construction, we develop a multi-term semi-tensor product singular value decomposition (MSTP-SVD), which integrates multiple orthogonal decomposition terms to significantly improve low-rank approximation accuracy compared with single-term schemes. To reduce the computational cost of multi-term modeling, we incorporate randomized projection and power iteration techniques into the MSTP-SVD framework, yielding an accelerated multi-term randomized semi-tensor product SVD (MRSTP-SVD) algorithm that achieves a balance between reconstruction accuracy and computational efficiency. Experiments on image and video compression and completion tasks demonstrate the effectiveness of the proposed method.

Xingchen Xiao (School of Mathematics and Statistics, Southwest University, Chongqing +13

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

Tensor Completion Tensor Decomposition Low-Rank Approximation Singular Value Decomposition

Sep 7, 2026cs.CV

Mind the Approximation: Fisher-Weighted SVD Compression for ViTs

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.

Moritz Thoma, Maximilian Groezinger, Maximilian Forstenhäusler +7

Technical University of Munich, DE · BMW Group, DE · University of Glasgow, GB +1

Efficient ViTs Low-Rank Compression Singular Value Decomposition Neural Network Compression

Aug 31, 2026math.ST

Compact and Infinite-Order Error Analysis for Null-Space SVD Estimation

We study null-space estimation from a noisy matrix. For a simple left null space, we first derive an exact compact expression for the error of the smallest left singular vector. We then give an all-order series for the SVD vector and projector, followed by compact and consistently truncated series forms for the fixed-realization empirical risk and conditional population generalization risk. The recursion extends to a multiple-dimensional null space by following the complete invariant subspace. The convergence radius is not inferred from an error plot: it is computed independently from the nearest complex exceptional point that joins a retained eigenvalue branch to its complement. A reduced-nullity experiment shows that moving this spectral boundary can increase the radius, although the improvement is not monotone in the retained nullity. For individually ordered null directions under Gaussian training with

τ\geq m

, we prove that the Wishart splitting matrix

W

gives a strict second-order empirical ranking. Gaussian averaging equalizes the leading generalization risks at both small and very large noise, while a column-swap theorem proves strict expected generalization ranking for an isotropic signal subspace. For unequal spikes, an exact population-overlap criterion and a simultaneous

99\%

Monte Carlo confidence certificate explain the observed intermediate ranking. A sixth-order risk correction improves the lower-crossover estimate in the reported experiment. This equal--ranked--equal phenomenon is a finite-sample diagnostic related to spectral mixing, but its tolerance crossings, the exceptional-point radius, and the asymptotic BBP threshold are three distinct quantities.

Xin Li, Jonathan Cohen, Rami Puzis

Random Matrix Theory Singular Value Decomposition

Jul 28, 2026cs.CV

Quasi-SVD: Learning a Lie-constrained matrix factorisation for real-time imaging

Singular Value Decomposition (SVD) underlies matrix factorisation tasks across many fields, with imaging applications demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in imaging pipelines. This study introduces a fully parallelized matrix factorization framework for GPUs by enforcing matrix orthogonality on left singular vectors via Lie-parametrised algebra and recovering the remaining components through soft constraints. This asymmetric constraint design enables an efficient parallel and provably valid decomposition, achieves high reconstruction fidelity and substantially accelerates computation relative to the exact SVD, with real-time throughput exceeding standard video frame rates. Performance is evaluated on multiple imaging tasks spanning complementary computational regimes: (1) spatio-temporal background subtraction for ultrasound localisation microscopy, requiring high-dimensional matrix separation, (2) Mueller matrix polarimetry for neurosurgical tissue characterisation, requiring massive batch processing of small matrices, and (3) an MNIST denoising benchmark at an intermediate scale with known ground truth. Across regimes and instruments, the proposed framework demonstrates robust domain transfer at various matrix scales, sufficient for live image-guided workflows that classical solvers cannot currently support in these settings. By prioritising downstream reconstruction fidelity over exact spectral recovery, the proposed SVD framework makes structured matrix factorisation practical for real-time processing.

Christopher Hahne

University of Bern 3008 Bern, Switzerland

Lie Group Methods Computational Imaging Singular Value Decomposition Matrix Factorization

Jul 3, 2026cs.LG

LACE-SVD: Loss-Aware SVD with Cumulative Error Correction for LLM Compression

The rapid growth in the parameter scale of large language models (LLMs) has created a strong demand for efficient compression techniques. As a hardware-agnostic and highly compatible approach, low-rank compression has been widely adopted to reduce both memory footprint and computational cost. However, existing SVD-based methods are still largely driven by local reconstruction objectives, overlooking two critical limitations: rank budgets are often allocated without explicitly considering layer-wise loss sensitivity, and local approximation errors can propagate and accumulate through the residual stream, leading to amplified global deviations from the original model. To address these issues, we propose LACE-SVD, a Loss-Aware SVD framework with Cumulative Error correction for LLM compression. LACE-SVD first estimates the calibration negative-log-likelihood increase induced by candidate layer-wise compression ratios and solves a budget-constrained allocation problem to assign rank budgets. It then refines the compressed model with closed-form local updates and introduces a propagation-aware correction for residual-stream output modules, reducing layer-output discrepancy as a proxy for cumulative error propagation. Experimental results demonstrate that at a high compression ratio (0.6), the WikiText-2 PPL of our method on LLaMA-7B (32.57) is significantly better than that of Dobi-SVD (46.18).

Zhuowen Liu, Longkun Hao, Shiyu Feng +3

Zhuowen Liu1 · Longkun Hao2 · Shiyu Feng3 +3

Model Compression LLM Compression Low-Rank Matrix Decomposition Low-Rank Compression Low-Rank Approximation Singular Value Decomposition

Jun 30, 2026stat.ML

CORA: Per-Slice Coherent Orthogonal Rotation for SVD-based Low-Rank Adaptation

Parameter-Efficient Fine-Tuning (PEFT) commonly adapts pretrained weights through low-rank updates, and recent methods further exploit the singular value decomposition (SVD) of the base weight for initialization or subspace selection. However, these methods do not explicitly preserve the coupled geometry between the pretrained left and right singular bases. Motivated by recent minimum-perturbation theory, which shows that stable finetuning follows a coherent SVD rotation in which a single orthogonal

Q

acts on both the left singular basis

U_0

and the right singular basis

V_0

, we prove a per-slice analogue: each row slice of

W_0

can be adapted by a shared orthogonal rotation

Q_i

on its left basis

U_i

and right basis

V_i

together with a diagonal spectrum shift. We implement this form as CORA (Coherent Orthogonal Rotation Adaptation), which applies per-slice orthogonal rotations and a per-layer diagonal scale to the rank-

r

SVD truncation of

W_0

. CORA uses

\tfrac{1}{2}m(r{-}1)

trainable parameters per linear layer, about

4{\times}

fewer than LoRA at the same rank. CORA outperforms LoRA, DoRA, PiSSA, and MiLoRA on commonsense reasoning and code generation while using about

8{\times}

fewer parameters.

Pengcheng Wang, Ziran Liu, Wei Wang +1

Purdue University · Shanghai Institute for Mathematics and Interdisciplinary Sciences / Fudan University · Futurewei Technologies, Inc.

LLM Fine-Tuning Low-Rank Matrix Decomposition Low-Rank Adaptation Singular Value Decomposition Parameter-Efficient Fine-Tuning

Jun 22, 2026cs.LG

SVD-Surgeon: Optimal Singular-Value Surgery for Large Language Model Compression

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.

Mahmoud Safari, Frank Hutter

University of Freiburg · Prior Labs · ELLIS Institute Tübingen

LLM Pruning LLM Compression Low-Rank Matrix Decomposition Low-Rank Compression Low-Rank Approximation Singular Value Decomposition+1

Jun 3, 2026stat.ML

Sparse Functional Singular Value Decomposition for Biclustering and Triclustering Longitudinal Data

Identifying subtypes of complex conditions, such as Inflammatory Bowel Disease (IBD), often requires capturing latent patterns in longitudinal omics data. However, these data are typically high-dimensional, sparsely sampled, and irregularly observed over time, posing substantial challenges for conventional (bi)clustering and functional data analysis methods. We propose Tri-SfSVD, a unified sparse functional Singular Value Decomposition framework for discovering biclusters and triclusters in longitudinal data. Unlike existing functional biclustering methods that rely on ad hoc imputation or enforce restrictive shape-homogeneity assumptions, Tri-SfSVD integrates continuous trajectory estimation with simultaneous subject, feature, and temporal selection within a single optimization framework. By imposing sparse penalties across subjects, variables, and temporal subregions, the proposed method works directly on observed data to uncover localized structures at the subject, subject-feature, and subject-feature-time levels. Extensive simulations demonstrate that Tri-SfSVD outperforms existing approaches in high-dimensional settings. Applied to IBD multi-omics data, the method identified three biclusters linking sample clusters with distinct IBD-related clinical characteristics to microbial pathway groups associated with specific bacterial taxa, providing interpretable subject-pathway associations for characterizing disease heterogeneity. Applied to multi-channel EEG data, the method identified three triclusters linking sample clusters with distinct alcohol-related phenotypes to localized brain activity patterns, including subgroup differences separated by temporal subregions within the same spatial region.

Yue Zhao, Thierry Chekouo, Sandra Safo

Division of Biostatistics and Health Data Science University of Minnesota 2221 University Avenue SE2026 Minneapolis, MN 55414, United States

Clustering Singular Value Decomposition

May 30, 2026cs.LG

LASER: Loss-Aware Singular-value Decomposition and Rank Allocation for Efficient Low-Precision Vision-Language Models

Vision-language models (VLMs) deliver strong multimodal reasoning capabilities, but their large computational cost and high parameter counts make deployment challenging on resource-constrained devices. Low-rank decomposition has emerged as a promising compression technique, yet existing methods often optimize local matrix reconstruction error, rely on uniform or heuristic rank allocation, and focus mainly on attention projections while leaving feed-forward networks underexplored. In this paper, we propose~\textit{LASER} (\textbf{L}oss-\textbf{A}ware \textbf{S}ingular-value d\textbf{E}composition and \textbf{R}ank allocation), a low-rank compression framework for efficient low-precision VLM inference. LASER derives a curvature-weighted SVD objective from a second-order approximation of the model loss and uses Kronecker-factored Fisher information to guide decomposition toward downstream performance rather than reconstruction alone. We further introduce a loss-aware cross-layer rank allocation strategy based on calibration gradients, enabling more effective parameter budgeting across layers. Finally, we extend low-rank compression to FFN layers through a hybrid scheme that combines SVD with quantization. The evaluation results show that LASER achieves more than

2.3\times

decoding speedup over previous work while preserving strong accuracy under low-precision inference.

Haiyu Wang, Yutong Wang, Leshu Li +2

Tandon School of Engineering, New York University · Courant Institute of Mathematical Sciences, New York University · Brookhaven National Laboratory

Vision-Language Models Efficient VLM Inference Low-Rank Matrix Decomposition Low-Rank Compression Singular Value Decomposition

May 27, 2026cs.LG

History-aware adaptive reduced-order models via incremental singular value decomposition

Reduced-order models (ROMs) can accelerate high-dimensional dynamical simulations, but their accuracy often deteriorates when online dynamics leave the regime represented by offline training data. We develop a projection-based adaptive ROM framework based on incremental singular value decomposition (iSVD), in which occasional full-order operator evaluations provide correction snapshots for online basis updates. The intrusive ROMs considered here are fully parameterized by the basis, so each update naturally propagates to reduced operators and hyper-reduction machinery. Through its evolving singular structure, iSVD retains an encoded history of the observed dynamics and is history-aware in this sense. We study the method on three nonlinear problems of increasing complexity: the one-dimensional viscous Burgers equation, the Sod shock tube, and a stiff one-dimensional ten-species rotating detonation engine (RDE). The Burgers problem is used to analyze the method and compare iSVD with alternative basis adaptation rules, showing that history-aware updates outperform instantaneous updates and that iSVD gives the strongest overall performance. The Sod and RDE cases demonstrate that these advantages persist in more challenging compressible-flow settings. For the RDE problem, the iSVD adaptive ROM improves upon the current state-of-the-art Direct adaptive ROM baseline in both predictive accuracy and computational efficiency. A cost analysis shows that the dominant online cost comes from interacting with the full-order model to obtain correction snapshots, while the iSVD update itself is negligible. These results identify iSVD as an effective mechanism for online learning of reduced subspaces and suggest a path toward ROMs that remain predictive over horizons several orders of magnitude longer than their initial training window.

Amirpasha Hedayat, Ali Mohaghegh, Laura Balzano +2

Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USA · Department of Aerospace Engineering, University of Kansas, Lawrence, KS, USA · Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA

Reduced-Order Modeling Singular Value Decomposition Computational Fluid Dynamics

May 27, 2026cs.LG

Efficient Pre-Training of LLMs through Truncated SVD Representations

LLM pretraining is extremely costly; therefore, parameter-efficient LLM architectures have recently emerged as a compelling research direction. One such promising approach is to represent the parameters as orthonormal low-rank weight matrices. However, maintaining orthonormality during training is computationally expensive, making it impractical. This paper presents the TSVD (Truncated Singular Value Decomposition) framework which efficiently maintains orthonormality through QR decomposition and caching. Furthermore, a spectral energy heuristic is introduced to select the rank of the resulting low-rank weight matrices. Empirical evaluations across model sizes show that TSVD matches or outperforms full-parameter baselines at a fraction of the compute cost. TSVD thus provides a scalable, computationally efficient foundation for LLM pretraining.

Kaivan Kamali, Kajetan Schweighofer, Hormoz Shahrzad +3

Cognizant AI Lab

Language Model Pretraining Low-Rank Matrix Decomposition Low-Rank Compression Low-Rank Approximation Singular Value Decomposition

May 19, 2026cs.LG

Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery

We introduce the

\star_G

tensor algebra, in which any finite group

G

defines the multiplication rule, making equivariance an intrinsic algebraic property rather than an architectural constraint. The framework rests on three machine-verified theoretical pillars: (i)~an Eckart-Young optimality guarantee for the

\star_G

-SVD: the first such result for symmetry-preserving tensor approximation, exact and polynomial-time; (ii)~a Kronecker factorization that composes multiple symmetries by replacing

F_G

with

F_{G_1} \otimes F_{G_2}

with no architectural redesign; and (iii)~~a 600-line Lean~~4 formalization of the

\star_G

algebra. The framework provides capabilities that equivariant neural networks (ENNs) structurally cannot: a closed-form per-irreducible-representation decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset. As a non-trivial empirical demonstration, decomposing QM9 molecular geometry over the chiral octahedral subgroup of SO(3) recovers the Wigner--Eckart selection rules of angular momentum from data alone, with no quantum mechanical input: scalar properties are A

_1

-dominated, dipole components are T

_1

-dominated, the isotropic polarizability is uniquely insensitive to

l\!=\!1

as the rank-2-trace decomposition

l\!=\!0 \oplus l\!=\!2

requires, and the T

_1

/A

_1

predictive-power ratio separates vector observables from scalar observables by a factor of five. On full QM9 (130{,}831 molecules),

\star_G

-SVD with ridge regression provides closed form predictions at

\sim50-90\times

fewer parameters than parameter-matched MLPs. Algebraic equivariance thus complements architectural equivariance not as a faster-better-cheaper alternative but as a different mathematical affordance: provably-optimal symmetry-preserving compression, per-irrep interpretability, and data-driven physical discovery.

Paulina Hoyos, Shashanka Ubaru, Dongsung Huh +5

UT Austin · IBM Research · Independent +2

Scientific Discovery Equivariant Neural Networks Tensor Decomposition Equivariant Representation Learning Singular Value Decomposition

May 10, 2026math.NA

Accelerating Power Method with Fast Sketching for Stronger Low-Rank Approximation

The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.

Shabarish Chenakkod, Michał Dereziński

University of Michigan, Ann Arbor, MI 48105.

Low-Rank Matrix Decomposition Randomized Sketching Low-Rank Approximation Spectral Methods Singular Value Decomposition

May 10, 2026math.OC

Phases of Muon: When Muon Eclipses SignSGD

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.

Elliot Paquette, Noah Marshall, Lucas Benigni +3

Mathematics and Statistics Department, McGill University · Mathematics and Statistics Department, Université de Montréal · Google DeepMind

Sign-Based Optimization Muon Optimizer Stochastic Optimization Convergence Singular Value Decomposition

May 8, 2026cs.LG

FlashSVD v1.5: Making Low-Rank Transformers Inference Actually Fast

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.

Wenhao Wu, Zishan Shao, Kangning Cui +5

Department of Electrical & Computer Engineering, Duke University · Department of Statistical Science, Duke University · Department of Computer Science, Wake Forest University +1

Efficient Transformer Inference LLM Compression LLM Inference Acceleration Low-Rank Compression Singular Value Decomposition

May 8, 2026math.NA

Sparse Random-Feature Neural Networks with Krylov-Based SVD for Singularly Perturbed ODE

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.

Kevin Kurian Thomas Vaidyan, Siddharth Rout

Institute of Applied Mathematics University of British Columbia Vancouver, BC V6T 1Z4

Neural PDE Solvers Singular Value Decomposition Krylov Subspace Methods

May 7, 2026cs.LG

A Generalized Singular Value Theory for Neural Networks

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.

Brian Charles Brown, Robert Bridges, David Grimsman +2

AI Sweden Gothenburg, Sweden

Neural Representation Geometry Geometric Representation Learning Singular Value Decomposition

Apr 18, 2026eess.IV

Structured 3D-SVD: A Practical Framework for the Compression and Reconstruction of Biological Volumetric Images

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.

Mario Aragonés Lozano, Oscar Romero, Antonio León

Communications Department, Universitat Politècnica de València, 46022 Valencia, Spain

Image Reconstruction Tensor Decomposition Singular Value Decomposition

Nov 20, 2025cs.LG

FairLRF: Achieving Fairness through Sparse Low Rank Factorization

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.

Yuanbo Guo, Jun Xia, Yiyu Shi

University of Notre Dame Notre Dame, Indiana, USA

Algorithmic Fairness Group Fairness Low-Rank Approximation Singular Value Decomposition Matrix Factorization

Jul 29, 2025stat.ML

Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration

Modern data analysis increasingly requires identifying shared latent structure across multiple high-dimensional datasets. A commonly used model assumes that the data matrices are noisy observations of low-rank matrices with a shared singular subspace. In this case, two primary methods have emerged for estimating this shared structure, which vary in how they integrate information across datasets. The first approach, termed Stack-SVD, concatenates all the datasets, and then performs a singular value decomposition (SVD). The second approach, termed SVD-Stack, first performs an SVD separately for each dataset, then aggregates the top singular vectors across these datasets, and finally computes a consensus amongst them. While these methods are widely used, they have not been rigorously studied in the proportional asymptotic regime, which is of great practical relevance in today's world of increasing data size and dimensionality. Consequently, it remains unclear when one method should be preferred over another. In this work, we derive exact expressions for the asymptotic performance and phase transitions of these two methods and develop optimal weighting schemes to further improve both methods. Our analysis reveals that while neither method uniformly dominates the other in the unweighted case, optimally weighted Stack-SVD dominates optimally weighted SVD-Stack when the low rank signal is fully shared across the datasets. We then analyze multiple, partially shared components per dataset and demonstrate that SVD-Stack can yield improved performance without requiring estimation of subspace alignment. Finally, we provide practical algorithms for estimating optimal weights from data, offering theoretical guidance for method selection in practical data integration problems. Extensive numerical simulations and semi-synthetic experiments on genomic data corroborate our theoretical findings.

Tavor Z. Baharav, Phillip B. Nicol, Rafael A. Irizarry +1

Eric and Wendy Schmidt Center, Broad Institute, Cambridge, MA, 02142, USA · Department of Data Science, Dana Farber Cancer Institute, Boston, MA, 02115, USA · Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, MA, 02115, USA

Low-Rank Approximation Singular Value Decomposition