Singular Value Decomposition

Recent work has shown that removing orthogonalization during training and applying it only at inference improves rotation estimation in deep learning, with empirical evidence favoring 9D representations with SVD projection. However, the theoretical understanding of why SVD orthogonalization specifically harms training, and why it should be preferred over Gram-Schmidt at inference, remains incomplete. We provide a detailed gradient analysis of SVD orthogonalization specialized to $3 \times 3$ matrices and $SO(3)$ projection. Our central result derives the exact spectrum of the SVD backward pass Jacobian: it has rank $3$ (matching the dimension of $SO(3)$) with nonzero singular values $2/(s_i + s_j)$ and condition number $κ= (s_1 + s_2)/(s_2 + s_3)$, creating quantifiable gradient distortion that is most severe when the predicted matrix is far from $SO(3)$ (e.g., early in training when $s_3 \approx 0$). We further show that even stabilized SVD gradients introduce gradient direction error, whereas removing SVD from the training loop avoids this tradeoff entirely. We also prove that the 6D Gram-Schmidt Jacobian has an asymmetric spectrum: its parameters receive unequal gradient signal, explaining why 9D parameterization is preferable. Together, these results provide the theoretical foundation for training with direct 9D regression and applying SVD projection only at inference.

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

The deployment of Large Language Models is constrained by the memory and bandwidth demands of static weights and dynamic Key-Value cache. SVD-based compression provides a hardware-friendly solution to reduce these costs. However, existing methods suffer from two key limitations: some are suboptimal in reconstruction error, while others are theoretically optimal but practically inefficient. In this paper, we propose Swift-SVD, an activation-aware, closed-form compression framework that simultaneously guarantees theoretical optimum, practical efficiency and numerical stability. Swift-SVD incrementally aggregates covariance of output activations given a batch of inputs and performs a single eigenvalue decomposition after aggregation, enabling training-free, fast, and optimal layer-wise low-rank approximation. We employ effective rank to analyze local layer-wise compressibility and design a dynamic rank allocation strategy that jointly accounts for local reconstruction loss and end-to-end layer importance. Extensive experiments across six LLMs and eight datasets demonstrate that Swift-SVD outperforms state-of-the-art baselines, achieving optimal compression accuracy while delivering 3-70X speedups in end-to-end compression time. Our code is available at https://github.com/hiahei/Swift-SVD.

While nowadays visual anomaly detection algorithms use deep neural networks to extract salient features from images, the high dimensionality of extracted features makes it difficult to apply those algorithms to large data with 1000s of images. To address this issue, we present an incremental dimension reduction algorithm to reduce the extracted features. While our algorithm essentially computes truncated singular value decomposition of these features, other than processing all vectors at once, our algorithm groups the vectors into batches. At each batch, our algorithm updates the truncated singular values and vectors that represent all visited vectors, and reduces each batch by its own singular values and vectors so they can be stored in the memory with low overhead. After processing all batches, we re-transform these batch-wise singular vectors to the space spanned by the singular vectors of all features. We show that our algorithm can accelerate the training of state-of-the-art anomaly detection algorithm with close accuracy.

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.

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.