COALA: Numerically Stable and Efficient Framework for Context-Aware Low-Rank Approximation

2025-07-10

AI Summary

Existing context-aware low-rank approximation methods rely on explicit computation and inversion of Gram matrices, which often causes numerical instability—leading to degraded approximation quality or singular solutions. Method: We propose an efficient, inversion-free framework grounded in stable matrix decompositions (e.g., QR or SVD), avoiding explicit Gram matrix construction via regularization and robust numerical algorithms. Contribution/Results: We theoretically establish convergence guarantees and derive rigorous error bounds. The method maintains high-accuracy approximation under challenging conditions—including memory constraints, data sparsity, and near-singular inputs—effectively preventing numerical degradation. Empirical evaluations demonstrate its efficiency and robustness in large-model compression and adapter-based fine-tuning tasks. By providing a numerically reliable foundation, our approach advances context-aware low-rank modeling with sound mathematical grounding.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Natural Language Processing: Safety and Robustness
• Computer Vision: Large Vision Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Recent studies suggest that context-aware low-rank approximation is a useful tool for compression and fine-tuning of modern large-scale neural networks. In this type of approximation, a norm is weighted by a matrix of input activations, significantly improving metrics over the unweighted case. Nevertheless, existing methods for neural networks suffer from numerical instabilities due to their reliance on classical formulas involving explicit Gram matrix computation and their subsequent inversion. We demonstrate that this can degrade the approximation quality or cause numerically singular matrices. To address these limitations, we propose a novel inversion-free regularized framework that is based entirely on stable decompositions and overcomes the numerical pitfalls of prior art. Our method can handle possible challenging scenarios: (1) when calibration matrices exceed GPU memory capacity, (2) when input activation matrices are nearly singular, and even (3) when insufficient data prevents unique approximation. For the latter, we prove that our solution converges to a desired approximation and derive explicit error bounds.

Problem

• Address numerical instability in low-rank approximation methods
• Handle large calibration matrices exceeding GPU memory
• Ensure stable approximation with nearly singular input matrices

Innovation

• Inversion-free regularized framework for stability
• Handles large matrices exceeding GPU memory
• Ensures convergence with explicit error bounds