Low-rank adaptation (LoRA) has become a popular parameter-efficient fine-tuning method for large language models. A key challenge in LoRA is how to determine the rank of each adaptation matrix, as rank directly controls its capacity and efficiency.

Existing adaptive-rank methods typically allocate ranks according to manually designed importance scores, which are not directly derived from an optimization objective. In this work, we propose $\ell_p$-LoRA, a principled rank-allocation method based on $\ell_p$ regularization with $0<p<1$, which is a classical sparsity-inducing technique in signal processing and statistics.

Specifically, we regularize the energy of each rank-one LoRA component, encouraging redundant components to vanish while preserving important ones. We derive the corresponding proximal subproblem and reduce the matrix optimization to a two-dimensional problem, leading to an implicit thresholding criterion for identifying redundant components.

Experiments on natural language understanding and question-answering tasks demonstrate that the proposed method achieves competitive performance with existing LoRA baselines.