  Low-Rank Adaptation (LoRA) is a widely used Parameter-Efficient Fine-Tuning
(PEFT) method that updates an initial weight matrix $W_0$ with a delta matrix
$\Delta W$ consisted by two low-rank matrices $A$ and $B$. A previous study
suggested that there is correlation between $W_0$ and $\Delta W$. In this
study, we aim to delve deeper into relationships between $W_0$ and low-rank
matrices $A$ and $B$ to further comprehend the behavior of LoRA. In particular,
we analyze a conversion matrix that transform $W_0$ 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."
