Abstract: While overparameterization in machine learning models offers great benefits in terms of optimization and generalization, it also leads to increased computational requirements as model sizes grow. In this work, we show that leveraging inherent low-dimensional structure within the model parameter updates, we can reap the benefits of overparameterization without the computational burden. In practice, we demonstrate the effectiveness of this approach for deep low-rank matrix completion as well as fine-tuning language models. For theory of deep overparameterized low-rank matrix recovery, we show that the learning dynamics of each weight matrix are confined to an invariant low-dimensional subspace. Consequently, we can construct and train compact, highly compressed factorizations possessing the same benefits as their overparameterized counterparts. For language model fine-tuning, we introduce a method called “Deep LoRA”, which improves the existing low-rank adaptation (LoRA) technique, leading to reduced overfitting and a simplified hyperparameter setup, all while maintaining comparable efficiency. The effectiveness of Deep LoRA is validated through its performance on natural language understanding tasks, particularly when fine-tuning with a limited number of samples.