Abstract: Current parameter-efficient fine-tuning (PEFT) methods build adapters widely agnostic of the context of downstream task to learn, or the context of important knowledge to maintain. As a result, there is often a performance gap compared to full-parameter fine-tuning, and meanwhile the fine-tuned model suffers from catastrophic forgetting of the pre-trained world knowledge. In this paper, we propose **CorDA**, a Context-oriented Decomposition Adaptation method that builds learnable **task-aware adapters** from weight decomposition oriented by the context of downstream task or the world knowledge to maintain. Concretely, we collect a few data samples, and perform singular value decomposition for each linear layer of a pre-trained LLM multiplied by the covariance matrix of the input activation using these samples. The inverse of the covariance matrix is multiplied with the decomposed components to reconstruct the original weights. By doing so, the context of the representative samples is captured through deciding the factorizing orientation. Our method enables two options, the **knowledge-preserved adaptation** and the **instruction-previewed adaptation**. For the former, we use question-answering samples to obtain the covariance matrices, and use the decomposed components with the smallest $r$ singular values to initialize a learnable adapter, with the others frozen such that the world knowledge is better preserved. For the latter, we use the instruction data from the fine-tuning task, such as math or coding, to orientate the decomposition and train the largest $r$ components that most correspond to the task to learn. We conduct extensive experiments on Math, Code, and Instruction Following tasks. Our knowledge-preserved adaptation not only achieves better performance than LoRA on fine-tuning tasks, but also mitigates the forgetting of world knowledge. Our instruction-previewed adaptation is able to further enhance the fine-tuning performance to be comparable with full fine-tuning, surpassing the state-of-the-art PEFT methods such as LoRA, DoRA, and PiSSA.