Foundation models have achieved tremendous success in different domains. However, their huge computation and storage complexity make these models difficult to fine-tune and also less applicable in practice. Recent study shows training in Fourier domain can be an effective fine-tuning method in terms of both model performance and number of training parameters. In this work, we propose to further reduce the complexity by the factorization through the product of interleaved circulant and diagonal matrices. In addition, we address the case of non-square fine-tuning weights by partitioning the circulant matrix into blocks. Our method avoids the construction of weight change matrix and utilizes 1D fast Fourier transform (FFT) instead of 2D FFT. Experimental results show that our method achieves similar or better performance across various tasks with much less floating-point operations (FLOPs) and the number of trainable parameters.

Low-rank adaptation (LoRA) has been demonstrated effective in reducing the trainable parameter number when fine-tuning a large foundation model (LLM). However, it still encounters computational and memory challenges when scaling to larger models or addressing more complex task adaptation. In this work, we introduce Sparse Spectrum Adaptation via Discrete Hartley Transformation (SSH), a novel approach that significantly reduces the number of trainable parameters while enhancing model performance. It selects the most informative spectral components across all layers, under the guidance of the initial weights after a discrete Hartley transformation (DHT). The lightweight inverse DHT then projects the spectrum back into the spatial domain for updates. Extensive experiments across both single-modality tasks such as language understanding and generation and multi-modality tasks such as video-text understanding demonstrate that SSH outperforms existing parameter-efficient fine-tuning (PEFT) methods while achieving substantial reductions in computational cost and memory requirements.

Low-rank adaptation (LoRA) has become a prevalent method for adapting pre-trained large language models to downstream tasks. However, the simple low-rank decomposition form may constrain the hypothesis space. To address this limitation, we introduce Location-aware Cosine Adaptation (LoCA), a novel frequency-domain parameter-efficient fine-tuning method based on inverse Discrete Cosine Transform (iDCT) with selective locations of learnable components. We begin with a comprehensive theoretical comparison between frequency-domain and low-rank decompositions for fine-tuning pre-trained large models. Our analysis reveals that frequency-domain approximation with carefully selected frequency components can surpass the expressivity of traditional low-rank-based methods. Furthermore, we demonstrate that iDCT offers a more efficient implementation compared to inverse Discrete Fourier Transform (iDFT), allowing for better selection and tuning of frequency components while maintaining equivalent expressivity to the optimal iDFT-based adaptation. By employing finite-difference approximation to estimate gradients for discrete locations of learnable coefficients on the DCT spectrum, LoCA dynamically selects the most informative frequency components during training. Experiments on diverse language and vision fine-tuning tasks demonstrate that LoCA offers enhanced parameter efficiency while maintains computational feasibility comparable to low-rank-based methods.

In the era of large language models, parameter-efficient fine-tuning (PEFT) has been extensively studied. However, these approaches usually rely on the space domain, which encounters storage challenges especially when handling extensive adaptations or larger models. The frequency domain, in contrast, is more effective in compressing trainable parameters while maintaining the expressive capability. In this paper, we propose a novel Selective Discrete Cosine Transformation (sDCTFT) fine-tuning scheme to push this frontier. Its general idea is to exploit the superior energy compaction and decorrelation properties of DCT to improve both model efficiency and accuracy. Specifically, it projects the weight change from the low-rank adaptation into the discrete cosine space. Then, the weight change is partitioned over different levels of the discrete cosine spectrum, and the most critical frequency components in each partition are selected. Extensive experiments on four benchmark datasets demonstrate the superior accuracy, reduced computational cost, and lower storage requirements of the proposed method over the prior arts. For instance, when performing instruction tuning on the LLaMA3.1-8B model, sDCTFT outperforms LoRA with just 0.05M trainable parameters compared to LoRA's 38.2M, and surpasses FourierFT with 30\% less trainable parameters. The source code will be publicly available.

Time Series Foundation Models (TSFMs) have recently garnered attention for their ability to model complex, large-scale time series data across domains such as retail, finance, and transportation. However, their application to sensitive, domain-specific fields like healthcare remains challenging, primarily due to the difficulty of fine-tuning these models for specialized, out-of-domain tasks with scarce publicly available datasets. In this work, we explore the use of Parameter-Efficient Fine-Tuning (PEFT) techniques to address these limitations, focusing on healthcare applications, particularly ICU vitals forecasting for sepsis patients. We introduce and evaluate two selective (BitFit and LayerNorm Tuning) and two additive (VeRA and FourierFT) PEFT techniques on multiple configurations of the Chronos TSFM for forecasting vital signs of sepsis patients. Our comparative analysis demonstrates that some of these PEFT methods outperform LoRA in terms of parameter efficiency and domain adaptation, establishing state-of-the-art (SOTA) results in ICU vital forecasting tasks. Interestingly, FourierFT applied to the Chronos (Tiny) variant surpasses the SOTA model while fine-tuning only 2,400 parameters compared to the 700K parameters of the benchmark.

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}.