An Overview of Low-Rank Structures in the Training and Adaptation of Large Models

Abstract: The rise of deep learning has revolutionized data processing and prediction in signal processing and machine learning, yet the substantial computational demands of training and deploying modern large-scale deep models present significant challenges, including high computational costs and energy consumption. Recent research has uncovered a widespread phenomenon in deep networks: the emergence of low-rank structures in weight matrices and learned representations during training. These implicit low-dimensional patterns provide valuable insights for improving the efficiency of training and fine-tuning large-scale models. Practical techniques inspired by this phenomenon, such as low-rank adaptation (LoRA) and training, enable significant reductions in computational cost while preserving model performance. In this paper, we present a comprehensive review of recent advances in exploiting low-rank structures for deep learning and shed light on their mathematical foundations. Mathematically, we present two complementary perspectives on understanding the low-rankness in deep networks: (i) the emergence of low-rank structures throughout the whole optimization dynamics of gradient and (ii) the implicit regularization effects that induce such low-rank structures at convergence. From a practical standpoint, studying the low-rank learning dynamics of gradient descent offers a mathematical foundation for understanding the effectiveness of LoRA in fine-tuning large-scale models and inspires parameter-efficient low-rank training strategies. Furthermore, the implicit low-rank regularization effect helps explain the success of various masked training approaches in deep neural networks, ranging from dropout to masked self-supervised learning.