Low-Rank Matrix Decomposition

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.

We study transfer learning for low-rank plus sparse matrix denoising when both ambient and latent dimensions grow. A data-rich source task provides subspace information for a higher-dimensional, data-scarce target with a small rank increment and localized sparse edits. We develop an anchored alternating-projection estimator that preserves transferred subspaces while learning target-specific coefficients, additional latent directions, and sparse modifications. We establish deterministic error bounds that separate target noise, representation growth, and source-estimation error, identifying improved rates through small structural increments and reduced noise within inherited subspaces. We further extend it to approximate anchoring quantifies robustness to subspace mismatch. We establish application-specific guarantees for Markov transition matrix estimation from single trajectories with dependent noise and for structured covariance estimation, with the latter analysis provided in the appendix. Experiments demonstrate consistent improvements over non-transfer baselines on synthetic data and an approximately 45% reduction in principal-angle error on a NYC taxi transition-estimation task with expanding state spaces.

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.