Low-Rank Matrix Decomposition

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.

Given a matrix $X$, and two ranks $r_1$ and $r_2$, the Hadamard decomposition (HD) looks for two low-rank matrices, $X_1$ of rank $r_1$ and $X_2$ of rank $r_2$, both of the same size as $X$, such that $X\approx X_1\circ X_2$, where $\circ$ is the Hadamard (element-wise) product. In most cases, HD is more expressive than standard low-rank approximations such as the truncated singular value decomposition (TSVD), as it can represent higher-rank matrices with the same number of parameters; this is because the rank of $X_1 \circ X_2$ is generically equal to $r_1 r_2$. In this paper, we first present some theoretical insights for HD, in particular a useful reformulation $X\approx WH^\top$ where $W$ and $H$ have $r_1 r_2$ columns and belong to certain manifolds. These allow us to develop three new algorithms for computing HD. The first one uses the representation $X\approx X_1\circ X_2$ and relies on the Manopt toolbox. The other two rely on the reformulation $X\approx WH^\top$: one is a block projected gradient method, and the other is a manifold-based gradient descent algorithm that does not require projection onto the feasible set. The last two algorithms are particularly effective for handling large sparse data. We also propose new initializations that allow us to improve the accuracy of the HD. We compare our algorithms and initialization strategies with the TSVD and with the state of the art. Numerical results show that the new methods are efficient and competitive on both synthetic and real data.

This paper studies the stability of low-rank implicit regularization in perturbed deep matrix factorization, where the target matrix is corrupted by a noise matrix. We first derive sufficient spectral conditions under which gradient descent exhibits a low-rank phase in the noiseless setting. These conditions show how the target spectrum, initialization, and step size jointly determine the existence of a nonempty low-rank interval. We then analyze the perturbed gradient descent dynamics, proving convergence guarantees and quantifying how the perturbation affects iteration complexity and eigenvalue recovery. Finally, we show that the low-rank phase persists under perturbation, with explicit dependence on the perturbation size. Numerical experiments support the theoretical findings.

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.

Modern matrix completion problems often involve heterogeneous data whose rows simultaneously belong to many meta-categories, such as demographic and age groups in recommendation systems, or region and recording session labels in neural electrophysiological experiments. Standard low-rank estimators impose a single global latent geometry, which can recover average structure but may smooth away subgroup-specific variation, especially when observations are unevenly distributed across groups. We introduce Group-Aware Matrix Estimation (GAME), a convex estimator for overlapping subgroup-wise low-rank matrix estimation. GAME regularizes category-specific submatrices through overlapping nuclear-norm penalties, allowing related groups to borrow information while preserving local latent structure in a shared coordinate system. We provide finite-sample guarantees for both reconstruction error and subgroup-specific subspace recovery, showing how performance depends on sampling density, subgroup rank, and overlap structure. Experiments on synthetic, recommendation, ecological, and neuroscience datasets show that GAME is most beneficial in structured missingness regimes, where subgroup-aware regularization improves both reconstruction accuracy and latent subspace fidelity. Across these benchmarks, GAME is competitive or best among global low-rank, side-information, and modern imputation baselines, with the largest gains when subgroups exhibit distinct low-rank structure.

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

Large language models deliver strong performance across language and reasoning tasks, but their storage and compute costs remain major barriers to deployment in resource-constrained and latency-sensitive settings. SVD-based post-training compression offers a hardware-agnostic way to reduce model size and improve inference efficiency through low-rank factorization. However, existing methods often rely on input-only whitening spaces, homogeneous rank allocation, or loss-agnostic allocation heuristics, limiting their ability to preserve model quality under aggressive compression. We propose Input-Output Whitened SVD (IO-SVD), a post-training compression method that forms a KL-aware double-sided whitening space for model weights. Using a second-order expansion of the KL loss over the top-K token probabilities, IO-SVD constructs an output-side metric that captures predictive sensitivity, while input whitening captures activation statistics. We further introduce an efficient heterogeneous rank-allocation strategy that scores whitened singular components using first-order calibration loss estimates and prunes the least sensitive components under a global budget. Inspired by prior work that combines SVD truncation with quantization, we improve hybrid SVD-quantization compression through loss-aware remapping, which selects low-rank factor rows for 8-bit quantization based on the predicted loss change incurred by quantizing them. Extensive experiments across diverse LLM and VLM families, and inference-time analysis shows that IO-SVD compresses LLMs with minimal performance degradation while delivering practical inference speedups. Code is available at https://github.com/mint-vu/IO-SVD.git

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.

Large language models (LLMs) have rapidly grown in scale, creating substantial memory and computational costs that hinder efficient deployment. Singular value decomposition (SVD) has emerged as an effective post-training compression technique, but existing SVD-based methods rely on static rank truncation, applying a fixed prefix of singular components to all inputs regardless of their diversity. We identify two limitations of this static design: the optimal rank varies across individual prompts, and the selected rank is sensitive to the choice of calibration set, leading to suboptimal performance across diverse inputs. To address these challenges, we propose $\textbf{PARSE}$, a post-training framework for $\textbf{P}$rompt-$\textbf{A}$ware $\textbf{R}$ank $\textbf{S}$election as $\textbf{E}$xperts in SVD-compressed LLMs. PARSE trains a linear router offline to perform prompt-aware rank selection, decoupling it from calibration information by supervising the router against dense-model outputs on a large-scale corpus. We further observe that rank-selection patterns are shared across semantically similar prompts and remain stable across decoding steps, allowing appropriate rank subsets to be served directly from a pattern cache at inference. Complemented by expert memory aggregation and kernel fusion for system-level efficiency, PARSE is orthogonal to existing SVD-based pipelines and consistently improves both model quality and inference efficiency. Integrated with four representative SVD-based methods, PARSE improves average task accuracy by up to 10% at a compression ratio of 0.6 on LLaMA-7B, and achieves up to 2.5 $\times$ prefill and 2.4 $\times$ decode speedup over native SVD execution.

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.

The massive computational costs of scaling modern deep learning architectures have driven the widespread use of parameter-efficient low-rank structures, such as LoRA and low-rank factorization. However, theoretical guarantees for their expressive power are less explored, often relying on restrictive priors like a pretrained base matrix, ReLU activations or non-verifiable singularity conditions. We first investigate the limits of neural networks constrained strictly to low-rank manifolds without pretrained dense priors. We demonstrate a theoretical paradox: while purely rank-1 layers can exactly interpolate arbitrary scalar datasets, they collapse for function approximations. To overcome this bottleneck without surrendering parameter efficiency, we introduce a unified \textit{Structural Correspondence} framework. We prove that augmenting low-rank layers with only a minimal sparse diagonal component, say a Diagonal plus Low-Rank (DLoR) structure, is sufficient to reach Universal Approximation. We show that any full-rank transformation can be exactly reconstructed using these DLoR components by trading off network width (additive decomposition) or depth (multiplicative decomposition). By tracking asymptotic Taylor remainders, we prove that DLoR neural networks fully restore the Universal Approximation Theorem for general activation functions. Finally, we establish that multiplicative depth provides superior parameter-to-expressivity scaling compared to additive width. Our results show that dense matrices and specific activation functions are not topological prerequisites for universal expressivity.

Low-rank decomposition is a compelling approach for compressing large language models, but its effectiveness hinges on selecting which singular-vector bases to retain for a target task. Existing methods such as Basel adapt singular-value coefficients on downstream data and prune bases with small re-learned magnitudes, a heuristic that can be misaligned with task performance because it ignores the local geometry of the loss landscape. We present Basis Selection with Importance (BSI), a principled low-rank compression framework that ranks and prunes bases by directly estimating the expected loss increase incurred when each basis is removed. BSI derives a derivative-based importance score from a second-order Taylor expansion of the task loss with respect to singular values, combining first-order sensitivity and second-order curvature to quantify pruning impact. To make this criterion practical for LLMs, we develop an efficient Hessian-diagonal estimator by adapting the Hutchinson randomized-probing method to loss curvature with symmetric parameter perturbations. We provide a comprehensive theoretical analysis, including loss-increase bounds under basis pruning, explicit propagation of Hessian-diagonal estimation error into these bounds, variance characterization tied to the Hessian spectrum, high-probability sample-complexity guarantees for achieving a target estimation accuracy, and guidance on perturbation intensity. Extensive experiments on mathematical reasoning benchmarks demonstrate that BSI consistently outperforms state-of-the-art low-rank decomposition baselines, with especially strong improvements under deep compression.

We consider the problems of computing the optimal rank-$1$ Hankel and Toeplitz-structured approximation of arbitrary matrices under $L_2$ and $L_1$-norm error. Such problems arise naturally in engineered systems, including the basic few-shot signal Direction-of-Arrival (DoA) estimation problem that is of importance to modern autonomous systems applications. We develop accurate and computationally efficient structured matrix decomposition algorithms for both formulations and then derive analytically grounded small-sample-support DoA estimators for practical sensing system deployments. The resulting estimators under the $L_2$ and $L_1$ norms are formally shown to be maximum-likelihood optimal under white Gaussian and Laplace noise, respectively. The estimators are further validated through extensive simulation studies and real-world data experiments in few-shot DoA inference.

We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate $I(δ;\,\cdot)$ whose boundary-inward property induces a monotone state parameter $δ_t$, thereby certifying that the trajectory is confined to a shrinking family of level sets. For certified initializations below the critical step size, this mechanism proves convergence to global minimizers. Above the critical step size, the same monotone-state mechanism instead leads to a balanced terminal regime; for a range of post-critical step sizes, the reduced dynamics exhibit period-2 behavior consistent with edge-of-stability phenomena. We further show that the scalar certificate is not an ad hoc algebraic construction: under structural axioms and a natural state-parameter normalization, it is uniquely determined by the monotonicity mechanism. Numerical experiments suggest that this state-dependent Lyapunov mechanism persists beyond the proved cases, including two-dimensional rank-1 approximation and quartic augmentations of scalar factorization.

Matrix-level low-rank compression is a promising way to reduce the cost of large language models, but running compression and evaluating the resulting models on language tasks can be prohibitively expensive. Can compression-induced degradation be predicted before committing to this compute? We systematically analyze the Qwen3 and Gemma3 model families across four representative low-rank compression methods: vanilla SVD, two ASVD variants, and SVD-LLM. We find that stable rank and information density, measured in bits per parameter, dominate performance degradation. The interaction term $γ\cdot \barρ_s$, defined as compression ratio times stable rank, is a robust predictor of accuracy degradation, achieving leave-one-out cross-validation Pearson correlations of $0.890$ for attention layers and $0.839$ for MLP layers. We provide theoretical intuition for why this predictor succeeds by connecting it to standard SVD truncation bounds and error composition mechanisms in transformer layers. These findings enable a predict-then-compress workflow: compute $γ\cdot \barρ_s$ from weights, estimate degradation, and invest compute only in desirable configurations.

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.

Foundation models have achieved remarkable success, yet their growing parameter counts pose significant computational and memory challenges. Low-rank factorization offers a promising route to reduce training and inference costs, but the community lacks a stable recipe for training models from scratch using exclusively low-rank weights while matching the performance of the dense model. We demonstrate that Large Language Models (LLMs) can be trained from scratch using exclusively low-rank factorized weights for all non-embedding matrices without auxiliary "full-rank" guidance required by prior methods. While native low-rank training often suffers from instability and loss spikes, we identify uncontrolled growth in the spectral norm (largest singular value) of the weight matrix update as the dominant factor. To address this, we introduce Spectron: Spectral renormalization with orthogonalization, which dynamically bounds the resultant weight updates based on the current spectral norms of the factors. Our method enables stable, end-to-end factorized training with negligible overhead. Finally, we establish compute-optimal scaling laws for natively low-rank transformers, demonstrating predictable power-law behavior and improved inference efficiency relative to dense models.

Activation-aware low-rank factorization techniques yield strong compression results but are generally confined to linear layers, while existing whitening-based theory typically makes an implicit full-rank assumption on activations. We introduce a layer representation framework that extends activation-aware factorization beyond linear layers, including standard and grouped convolutions. Within this framework, our whitening-based formulation is more general than prior ones, naturally covering rank-deficient activations, and yields an optimal low-rank projection that attains the reconstruction error of the best low-rank approximation to layer activations. The resulting singular spectrum provides a closed-form per-layer distortion proxy, which we use to allocate per-layer ranks under explicit FLOP or parameter-count budgets via a Lagrangian relaxation with negligible overhead. Together, these components form BALF, an end-to-end pipeline for efficient vision model compression. Across CNNs and vision transformers on CIFAR-10 and ImageNet-1K, BALF generally achieves higher accuracy than SVD-based factorization baselines at matched FLOP or parameter count targets and remains competitive with other fine-tuning-free compression techniques.