A Single Linear Layer Yields Task-Adapted Low-Rank Matrices

Low-Rank Adaptation (LoRA) is a widely used Parameter-Efficient Fine-Tuning (PEFT) method that updates an initial weight matrix $W0$ with a delta matrix $\Delta W$ consisted by two low-rank matrices $A$ and $B$. A previous study suggested that there is correlation between $W0$ and $\Delta W$. In this study, we aim to delve deeper into relationships between $W0$ and low-rank matrices $A$ and $B$ to further comprehend the behavior of LoRA. In particular, we analyze a conversion matrix that transform $W0$ into low-rank matrices, which encapsulates information about the relationships. Our analysis reveals that the conversion matrices are similar across each layer. Inspired by these findings, we hypothesize that a single linear layer, which takes each layer's $W_0$ as input, can yield task-adapted low-rank matrices. To confirm this hypothesis, we devise a method named Conditionally Parameterized LoRA (CondLoRA) that updates initial weight matrices with low-rank matrices derived from a single linear layer. Our empirical results show that CondLoRA maintains a performance on par with LoRA, despite the fact that the trainable parameters of CondLoRA are fewer than those of LoRA. Therefore, we conclude that "a single linear layer yields task-adapted low-rank matrices."