Parameter-Efficient Fine-Tuning (PEFT) methods, such as Low-Rank Adaptation (LoRA), have significantly reduced the number of trainable parameters needed in fine-tuning large language models (LLMs). The developments of LoRA-style adapters have considered two main directions: (1) enhancing model expressivity with high-rank adapters, and (2) aiming for further parameter reduction, as exemplified by vector-based methods. However, these approaches come with a trade-off, as achieving the expressivity of high-rank weight updates typically comes at the cost of sacrificing the extreme parameter efficiency offered by vector-based techniques. To address this issue, we propose a vector-based random Tensor network for high-Rank Adaptation (TeRA), a novel PEFT method that achieves high-rank weight updates while retaining the parameter efficiency of vector-based PEFT adapters. This is achieved by parametrizing the tensorized weight update matrix as a Tucker-like tensor network (TN), whereby large randomly initialized factors are frozen and shared across layers, while only small layer-specific scaling vectors, corresponding to diagonal entries of factor matrices, are trained. Comprehensive experiments demonstrate that TeRA matches or even outperforms existing high-rank adapters, while requiring as few trainable parameters as vector-based methods. Theoretical analysis and ablation studies validate the effectiveness of the proposed TeRA method. The code is available at https://github.com/guyuxuan9/TeRA.

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