  Low-rank adaptation~(LoRA) has recently gained much interest in fine-tuning
foundation models. It effectively reduces the number of trainable parameters by
incorporating low-rank matrices $A$ and $B$ to represent the weight change,
i.e., $\Delta W=BA$. Despite LoRA's progress, it faces storage challenges when
handling extensive customization adaptations or larger base models. In this
work, we aim to further compress trainable parameters by enjoying the powerful
expressiveness of the Fourier transform. Specifically, we introduce FourierFT,
which treats $\Delta W$ as a matrix in the spatial domain and learns only a
small fraction of its spectral coefficients. With the trained spectral
coefficients, we implement the inverse discrete Fourier transform to recover
$\Delta W$. Empirically, our FourierFT method shows comparable or better
performance with fewer parameters than LoRA on various tasks, including natural
language understanding, natural language generation, instruction tuning, and
image classification. For example, when performing instruction tuning on the
LLaMA2-7B model, FourierFT surpasses LoRA with only 0.064M trainable
parameters, compared to LoRA's 33.5M. Our code is released at
\url{https://github.com/Chaos96/fourierft}.
